Chapter VI
Products and Quotients
1. Introduction
In Chapter III we defined the product of two topological spaces \ and ] and developed some simple
properties of the product. (See Examples III.5.10-5.12 and Exercise IIIE20.) Using simple induction,
that work generalized easy to finite products. It turns out that looking at infinite products leads to
some very nice theorems. For example, infinite products will eventually help us decide which
topological spaces are metrizable.
2. Infinite Products and the Product Topology
The set \ ‚ ] was defined as ÖÐBß CÑ À B − \ ß C − ] ×. How can we define an “infinite product” set
\ œ Ö\α À α − E×? Informally, we want to say something like
\ œ Ö\α À α − E× œ ÖÐBα Ñ À Bα − \α ×
so that a point B in the product consists of “coordinates Bα ” chosen from the \α 's. But what exactly
does a symbol like B œ ÐBα Ñ mean if there are “uncountably many coordinates?”
We can get an idea by first thinking about a countable product. For sets \" ß \# ß ÞÞÞß \8 ,... we can
informally define the product set as a certain set of sequences: \ œ ∞
8œ" \8 œ ÖÐB8 Ñ À B8 − \8 ×.
But if we want to be careful about set theory, then a legal definition of \ should have the form
\ œ ÖÐB8 Ñ − Y À B8 − \8 ×. From what “pre-existing set” Y will the sequences in \ be chosen? The
answer is easy: we write

∞
\ œ ∞
8œ" \8 œ ÖB − Ð 8œ" \8 Ñ À BÐ8Ñ œ B8 − \8 ×Þ
Thus the elements of ∞
8œ" \8 are certain functions (sequences) defined on the index set . This idea
generalizes naturally to any productÞ
Definition 2.1
Let Ö\α À α − E× be a collection of sets. We define the product set
\ œ Ö\α À α − E× œ ÖB − \α E À BÐαÑ − \α ×. The \α 's are called the factors of \ . For each
α, the function 1α À Ö\α À α − E× Ä \α defined by 1α ÐBÑ œ Bα is called the αth -projection map.
For B − \ , we write more informally Bα œ BÐαÑ œ the αth -coordinate of B and write B œ ÐBα ).
Note: the index set E might not be ordered. So, even though we use the informal notation B œ ÐBα Ñ,
such phrases as “the first coordinate of B,” “the next coordinate in B after Bα ,” and “the coordinate
in B preceding Bα ” may not make sense. The notation ÐBα Ñ is handy but can lead you into errors if
you're not careful.
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A point B in Ö\α À α − E× is, by definition, a function that “chooses” a coordinate Bα from each set
in the collection Ö\α À α − E×. To say that such a “choice function” must exist if all the \α 's are
nonempty is precisely the Axiom of Choice. (See the discussion following Theorem I.6.8.)
Theorem 2.2 The Axiom of Choice (AC) is equivalent to the statement that every product of
nonempty sets is nonempty.
Note: In ZF set theory, certain special products can be shown to be nonempty without using AC.


For example, if \8 œ ß then ∞
8œ" \8 œ ÖB −  À BÐ8Ñ − × œ  . Without using AC, we can
precisely describe a point ( œ function) in the product  for example, the identity function
3 œ ÖÐ7ß 8Ñ −  ‚  À 7 œ 8×  so  œ ∞
8œ" \8 Á g.
We will often write Ö\α À α − E× as α−E \α . If the indexing set E is clearly understood, we may
simply write  \α .
Example 2.3
1) If E œ g, then Ö\α À α − E× œ ÖÐα−E \α ÑE À BÐαÑ − \α × œ Ög×Þ
2) Suppose \α! œ g for some α! − E. Then α−E \α œ ÖÐα−E \α ÑE À BÐαÑ − \α ×Þ
Since BÐα! Ñ − \α! is impossible, αα−E \α œ g.
3) Strictly speaking, we have two different definitions for a finite product \" ‚ \# :
i) \" ‚ \# œ ÖÐB" ß B# Ñ À B" − \" ß B# − \# ×
(a set of ordered pairs)
ii) \" ‚ \# œ ÖB − Ð\" ∪ \# ÑÖ"ß#× À BÐ3Ñ − \3 ×
(a set of functions)
But there is an obvious way to identify these two sets: the ordered pair ÐB" ß B# Ñ corresponds to the
function B œ ÖÐ"ß B" Ñß Ð#ß B# Ñ× − Ð\" ∪ \# ÑÖ"ß#× Þ

∞
4) If E œ , then Ö\8 À 8 − × œ ∞
8œ" \8 œ ÖB − Ð 8œ" \8 Ñ À B8 − \8 ×
œ ÖÐB" ß B# ß ÞÞÞß B8 ß ÞÞÞß Ñ À B8 − \8 × œ the set of all sequences ÐB8 Ñ where B8 − \8 Þ
5) Suppose the \α 's are identical, say \α œ ] for all α − E. Then Ö\α À α − E×
œ ÖB − Ðα−E \α ÑE À BÐαÑ − \α × œ ÖB − ] E À Bα − ] × œ ] E . If lEl œ 7, we will sometimes
write this product simply as ] 7 œ “the product of 7 copies of ] ” because the number of factors 7 is
often important while the specific index set E is not.
Now that we have a definition of the set Ö\α À α − E×, we can think about a product topology. We
begin by recalling the definition and a few basic facts about the “weak topology.” (See Example
III.8.6.)
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Definition 2.4 Let \ be a set. For each α − E, suppose Ð\α ß gα Ñ is a topological space and that
0α À \ Ä \α . The weak topology on \ generated by the collection Y œ Ö0α À α − E× is the smallest
topology on \ that makes all the 0α 's continuous.
Certainly, there is at least one topology on \ that makes all the 0α 's continuous: the discrete topology.
Since he intersection of a collection of topologies on \ is a topology (why? ), the weak topology
exists  we can describe it “from the top down” as {g : g is a topology on \ making all the 0α 's
continuous×Þ
However, this slick description of the weak topology doesn't give a useful description of what sets are
open. Usually it is more useful to describe the weak topology on \ “from the bottom up.” to make all
the 0α 's continuous it is necessary and sufficient that
for each α − E and for each open set Yα © \α , the set 0α" ÒYα Ó must be open.
Therefore the weak topology g is the smallest topology that contains all such sets 0α" ÒYα Ó and that is
the topology for which Æ œ Ö0α" ÒYα Ó À α − Eß Yα open in \α × is a subbase. (See Example III.8.6.)
Therefore a base for the weak topology consists of all finite intersections of sets from Æ. A typical
basic open set has form 0α"
ÒYα" ] ∩ 0α"
ÒYα# Ó ∩ ... ∩ 0α"
ÒYα8 Ó where each α3 − E and each Yα3 is
"
#
8
open in \α3 . To cut down on symbols, we will use a special notation for these subbasic and basic open
sets:
We will write  Yα  for 0α" ÒYα ÓÞ
A typical basic open set is then Y œ 0α"
ÒYα" ] ∩ 0α"
ÒYα# Ó ∩ ... ∩ 0α"
ÒYα8 Ó
"
#
8
which we further abbreviate as Y œ  Yα" ß Yα# ß ÞÞÞß Yα8 
So B − Y œ  Yα" ß Yα# ß ÞÞÞYα8  iff 0α3 ÐBÑ − Yα3 for each 3 œ "ß ÞÞÞß 8Þ
This notation is not standard but it should be because it's very handy. You should verify that to get a
base for the weak topology g on \ , it is sufficient to use only the sets  Yα" ß Yα# ß ÞÞÞß Yα8  where
each Yα3 is a basic (or even subbasic) open set in \α3 .
Example 2.5 Suppose Ð\ß g Ñ is any topological space and E © \Þ Let 3 À E Ä \ be the inclusion
map 3Ð+Ñ œ +Þ Then the subspace topology on E is the same as the weak topology generated by
Y œ Ö3×Þ To see this, just note that a base for the weak topology is Ö3" ÒY Ó À Y open in \× and that
3" ÒY Ó œ Y ∩ E which is a typical open set in the subspace topology.
The following theorem tells us that a map 0 into a space \ with the weak topology is continuous iff
each composition 0α ‰ 0 is continuous.
Theorem 2.6 For α − E, suppose 0α À \ Ä Ð\α ß gα Ñ and that \ has the weak topology generated by
the functions 0α . Let ^ be a topological space and 0 À ^ Ä \ . Then 0 is continuous iff
0α ‰ 0 À ^ Ä \α is continuous for every α.
Proof If 0 is continuous, then each composition 0α ‰ 0 is continuous. Conversely, suppose each
0α ‰ 0 is continuous. To show that 0 is continuous, it is sufficient to show that 0 " ÒZ Ó is open in ^
whenever Z is a subbasic open set in \ (why? ). So let Z œ  Yα  with Yα open in \α . Then
0 " ÒZ Ó œ 0 " Ò0α" ÒYα ÓÓ œ Ð0α ‰ 0 Ñ" ÒYα Ó  which is open since 0α ‰ 0 is continuous. ñ
241
Definition 2.7 For each α − Eß let Ð\α ß gα Ñ be a topological space. The product topology g on the
set \α is the weak topology generated by the collection of projection maps Y œ Ö1α À α − E×Þ
The product topology is sometimes called the “Tychonoff topology.” We always assume that a
product space \α has the product topology unless something else is explicitly stated.
Since the product topology is a weak topology, a subbase for the product topology consists of all sets
of form  Yα  œ 1α" ÒYα Ó, where α − E and Yα is open in \αÞ A base consists of all finite
intersections of such sets À
 Yα"  ∩ ÞÞÞ ∩  Yα8  œ 1α"" ÒYα" ] ∩ ... ∩ 1α"8 ÒYα8 Ó
œ ÖB − \α À Bα3 − Yα3 for each 3 œ "ß ÞÞÞß 8×
œ α−E Yα where Yα œ \α for α Á α" ß α# ß ÞÞÞß α8 (*)
(It is sufficient to use only Yα ' s which are basic (or even subbasic) open sets in \α Þ Why?)
A basic open set in \α “depends on only finitely many coordinates” in the following sense:
B −  Yα" ß Yα# ß ÞÞÞß Yα8  iff B satisfies the finitely many restrictions Bα3 − Yα3 Ð3 œ "ß ÞÞÞß 8Ñ.
Since (basic) open sets containing B are the “standard” for measuring closeness to B, we can
say, roughly, that in the product topology “closeness depends on only finitely many
coordinates.”
For a finite index set E œ Ö"ß #ß ÞÞÞß 8×, the product topology on α−E \α is just the one for which the
basic open sets are the open boxes α−E Yα :
α−E Yα œ 83œ" Y3 œ  Yα" ß Yα# ß ÞÞÞß Yα8  œ Y" ‚ Y# ‚ ÞÞÞ ‚ Y8
Then condition (*) that “Yα œ \α for all but finitely many α' s” is satisfied automatically so, for finite
products the product topology is exactly as we defined it in Chapter III.
Definition 2.7 is probably not what was expected. Someone's “first guess” would more likely be to
use all “boxes” α−E Yα (Yα open in \α Ñ as a base in defining the product topology. But in Definition
2.7, a “box” of the form α−E Yα might not be open because (*) may not hold. It is perfectly possible,
of course, to define a different topology on the set α−E \α using all boxes of the form α−E Yα as a
base. This alternate topology on α−E \α is called the box topology. But we will see that our
definition of the product topology is the “right” definition to use. (Of course, the box topology and the
product topology coincide for finite products.)
Theorem 2.8 Each projection map 1α À ÖÖ\α À α − E× Ä \α is continuous and open.
If Ö\α À α − E× Á g, each 1α is onto. A function 0 À ^ Ä \α is continuous iff
1α ‰ 0 À ^ Ä \α is continuous for every α.
Proof By definition, the product topology is one that makes all the 1α 's continuous. and clearly, each
1α is onto if the product is nonempty. To show that 1α is open, it is sufficient to show that the image
of a basic open set Y œ  Yα" ß Yα# ß ÞÞÞß Yα8  is open. This is clearly true if Y œ g.
242
For Y Á g, we get 1α ÒY Ó œ 1α Ò  Yα" ß Yα# ß ÞÞÞß Yα8  Ó œ 
Yα 3
\α
if α œ α3
..
if α Á α" ß ÞÞÞß α8
Either way, 1α Ò  Yα" ß Yα# ß ÞÞÞß Yα8  Ó is open.
Because the product has the weak topology generated by the projections, Theorem 2.6 gives
that 0 is continuous iff each composition 1α ‰ 0 is continuous. ñ
Generally, projection maps are not closed. For example, if J œ ÖÐBß CÑ − ‘# À C œ B" ß B  !×,
then 1" ÒJ Ó œ the projection of J on the B-axis ‘ is not closed in ‘Þ
Example 2.9
1) A subbasic open set in ‘ ‚ ‘ has form 1"" ÒY Ó œ Y ‚ ‘ or 1#" ÒZ Ó œ ‘ ‚ Z , where Y and
Z are open in ‘Þ (We get a subbase even if we restrict ourselves just to using basic open intervals
Y ß Z in ‘Þ). So basic open sets have form ÐY ‚ ‘Ñ ∩ Б ‚ Z Ñ œ Y ‚ Z . Therefore the product
topology on ‘ ‚ ‘ is the usual topology on ‘# .
The function 0 À ‘ Ä ‘ ‚ ‘ given by 0 Ð>Ñ œ Ð># ß sin ># Ñ is continuous because the
compositions Ð1" ‰ 0 ÑÐ>Ñ œ ># and Ð1# ‰ 0 ÑÐ>Ñ œ sin ># are both continuous functions from ‘ to ‘.
2) Let \ œ i! œ “the product of countably many copies of .” A base for the product
topology consists of all sets Y œ  Y8 where finitely many Y8 's are singletons and all the others are
equal to Þ Each Y is infinite (in fact lY l œ -  why?), so every nonempty open set is infinite. In
particular, \ has no isolated pointsÞ More generally, an infinite product of (nonempty) discrete spaces
is not discrete.
The box topology on \ is the discrete topology. For each point B œ Ð5" ß 5# ß ÞÞÞß 58 , ...Ñ − \ ,
the set ÖB× œ ∞
8œ" Ö58 × is open in the box topology. (For a finite product, the box and product
topologies care the same: a finite product of discrete spaces is discrete.)
3) Let ‘‘ œ Ö\< À < − ‘×, where \< œ ‘. Each point 0 in ‘‘ is a function 0 À ‘ Ä ‘ and
1< Ð0 Ñ œ 0 Ð<ÑÞ A basic open set containing 0 is Y œ  Y<" ß Y<# ß ÞÞÞß Y<8  ß where the Y<3 's are basic
open in ‘ and 0 Ð<3 Ñ − Y<3 Þ
If each Y<3 is an open interval Ð0 Ð<3 Ñ  %3 ß 0 Ð<3 Ñ  %3 Ñ, then
Y œ Ö1 − ‘‘ À l 1Ð<3 Ñ  0 Ð<3 Ñl  %3 ß 3 œ "ß ÞÞÞß 8×. In other words, 1 − Y iff the vertical distance in
‘# between the graphs of 1 and 0 is  %3 at each of the finitely many points <3 .
Why is the product topology the “correct” topology for set  \α ? Of course there is no “right”
answer, but a few examples should make it seem a good choice.
Example 2.10
1) Our intuition can completely comprehend only finite objects; we always run risks when
we apply it to infinite collections. For finite products, the product and box topologies are
exactly the same, and our intuition gives us no solid reason to prefer one over the other in the
infinite case.
2) (See Example II.2.6.6 and Exercise II.E10)
Let L be the “Hilbert cube”
"
"
"
∞
Ò!ß
Ó
œ
Ò!ß
"Ó
‚
Ò!ß
Ó
‚
ÞÞÞ
‚
Ò!ß
Ó
‚
ÞÞÞ
©
j
,
where
j# has its usual metric, ..
#
8œ"
8
#
8
Suppose Bß C − L and let %  !. What condition on C will guarantee that .ÐCß BÑ  % ?
243
Pick R so that 
∞
3œR "
"
3#

%#
#Þ
If ÐB3  C3 Ñ# 
%#
#R
.ÐBß CÑ œ Ð ÐB3  C3 Ñ#   ÐB3  C3 Ñ# Ñ"Î#  ÐR †
R
3œ"
∞
3œR "
%#
#R
for each 3 œ "ß ÞÞÞß R , then we have

%# "Î#
#Ñ
œ %Þ
This is the natural
metric topology on the product L , and we see here that we can make “C close to B” by
requiring “closeness” in just finitely many coordinates "ß ÞÞÞß R Þ This is just what the product
topology does and, in fact, the product topology on L turns out to be the topology g. Þ.
In Ð!ß "Ñ we can see a similar phenomenon quite clearly: for two points B œ !ÞB" ÞÞÞB8 ÞÞÞ and
C œ !ÞC" ÞÞÞC8ÞÞÞ ß “C will be close to B” if C and B agree in, say, the first 8 decimal places.
Roughly speaking, “closeness” depends on only “finitely many coordinates.”
A handy “rule of thumb” that has proved true every time I've used it is that if a topology on a
product set makes “closeness” depend on only a finite number of coordinates, then that
topology is the product topology.
3) The bottom line: a mathematical definition justifies itself by the fruit it bears. The
definition of the product topology will lead to some beautiful theorems. For example, we will
see that compact Hausdorff spaces are (topologically) nothing other than the closed subspaces
of cubes Ò!ß "Ó7 with the product topology (7 may be infinite). For the time being, you will
need to accept that things work out nicely down the road, and that by contrast, the box
topology turns out to be rather ill-behaved. (See Exercise E11.)
As a simple example of nice behavior, the following theorem is exactly what one would hope for  and
the proof depends on having the “correct” definition for the product topology. The theorem says that
convergence of sequences in a product is “coordinatewise convergence” : that is, in a product,
ÐB8 Ñ Ä B iff for all α , the α>2 coordinate of B8 converges (in \α Ñ to the α>2 coordinate of B. The
product topology is sometimes called the “topology of coordinatewise convergence.”
Theorem 2.11 Suppose ÐB8 Ñ is a sequence in \ œ Ö\α À α − E×. Then ÐB8 Ñ Ä B − \ iff
(1α ÐB8 Ñ) Ä 1α ÐBÑ in \α for all α − E.
Proof If ÐB8 Ñ Ä B, then (1α ÐB8 Ñ) Ä 1α ÐBÑ because each 1α is continuous.
Conversely, suppose (1α ÐB8 Ñ) Ä 1α ÐBÑ in \α for each α and consider any basic open let
Y œ  Yα" ß Yα# ß ÞÞÞß Yα5  that contains B œ ÐBα Ñ. For each 3 œ "ß ÞÞÞß 5 , we have Bα3 − Yα3 . Since
(1α3 ÐB8 Ñ) Ä 1α 3 ÐBÑ, we have 1α3 ÐB8 Ñ − Yα3 for 8 some R3 . If R œ max ÖR" ß ÞÞÞß R5 ×, then for
8 R we have 1α3 ÐB8 Ñ − Yα3 for every 3 œ "ß ÞÞÞß 5 . This means that B8 − Y for 8 R , so
ÐB8 Ñ Ä B. ñ
In the proof, R is the max of a finite set. If \ has the box topology, the basic open set Y œ Yα
might involve infinitely many open sets Yα Á \α . For each such α, we could pick Rα , just as in the
proof. But the set of all Rα 's might not have a max R so the proof would collapse. Can you create a
specific example with the box topology where this happens?
Example 2.12 Consider ‘‘ œ Ö\< À < − ‘×, where \< œ ‘. Each point 0 in the product is a
function 0 À ‘ Ä ‘Þ Suppose that Ð08 Ñ is a sequence of points in ‘‘ . By Theorem 2.11, Ð08 Ñ Ä 0
iff Ð08 Ð<ÑÑ Ä 0 Ð<Ñ for each < − ‘. With the product topology, convergence of a sequence of functions
in ‘‘ is called (in analysis) pointwise convergence.
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Question: if ‘‘ is given the box topology, is convergence of a sequence Ð08 Ñ simply
uniform convergence (as defined in analysis)?
The following “theorem” is stated loosely. You can easily create variations. Any reasonable version
of the statement is probably true.
“Theorem 2.13” Topological products are associative in any “reasonable” sense: for example, if
E œ F ∪ G where F and G are disjoint indexing sets, then
Ö\α À α − E× ¶ Ö\" À " − F× ‚ Ö\# À # − G×
“Proof” A point B − Ö\α À α − E× is a function B À E Ä α−E \α . Define
0 À Ö\α À α − E× Ä Ö\" À " − F× ‚ Ö\# À # − G× by 0 ÐBÑ œ ÐBlFß BlGÑ.
Clearly 0 is one-to-one and onto.
For convenience, let ] œ Ö\" À " − F× and ^ œ Ö\# À # − G× so that
1" ‰ 0 À Ö\α À α − E× Ä ] and 1# ‰ 0 À Ö\α À α − E× Ä ^Þ
0 is a mapping into a product Y ‚ ^ , so 0 is continuous iff 1" ‰ 0 and 1# ‰ 0 are both continuous.
But 1" ‰ 0 À Ö\α À α − E× Ä ] œ Ö\" À − F× is also a map into a productß so 1" ‰ 0 is
continuous iff 1" ‰ 1" ‰ 0 À Ö\α À α − E× Ä \" is continuous for all " − FÞ
But 1" ‰ 1" ‰ 0 œ 1" À Ö\α À α − E× Ä \" which is continuous.
The proof that 1# ‰ 0 is continuous is completely similar.
0 " À ] ‚ ^ Ä Ö\α À α − E× is given by B œ 0 " ÐCß DÑ œ C ∪ D (the union of two functions) and
0 " is continuous iff 1α ‰ 0 " À ] ‚ ^ Ä \α is continuous for each α − E œ F ∪ GÞ
Suppose α − F: then
1α ‰ 0 " ÐCß DÑ œ 1α ÐBÑ, where B œ C ∪ DÞ
Since α − Fß Bα
"
œ Ð1α ‰ 1" ÑÐCß DÑß so 1α ‰ 0 œ 1α ‰ 1" , which is continuous. The case where α − G is completely
similar.
Therefore 0 is a homeomorphism. ñ
The question of topological commutativity for products only makes sense when the index set E is
ordered in some way. Even in that case, if we view a product as a collection of functions, the question
of commutativity is irrelevant  the question reduces to the fact that set theoretic unions are
commutative. For example, \" ‚ \# œ ÖB − Ð\" ∪ \# ÑÖ"ß#× À BÐ3Ñ − \3 for 3 œ "ß #×
œ ÖB − Ð\# ∪ \" ÑÖ"ß#× À BÐ3Ñ − \3 for 3 œ "ß #× œ \# ‚ \" Þ
So viewed as sets of functions, \" ‚ \# and \# ‚ \" are exactly the same set! The same observation
applies to any product viewed as a collection of functions.
But we might look at an ordered product in another way: for example, thinking of \" ‚ \# and
\# ‚ \" as sets of ordered pairs. Then generally \" ‚ \# Á \# ‚ \" . From that point of view, the
topological spaces \1 ‚ \2 and \2 ‚ \1 are not literally identical, but there is a homeomorphism
between them: 0 ÐB" ß B# Ñ œ ÐB# ß B" ÑÞ So the products are still topologically identical. We can make a
similar argument whenever any ordered product is “commuted” by permuting the index set.
The general rule of thumb is that “whenever it makes sense, topological products are commutative.”
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Exercise 2.14 Recall that for a space \ and cardinal 7, \ 7 denotes the product of 7 copies of the
space \ . Prove that Ð\ 7 Ñ8 is homeomorphic to \ 78 . (Hint: Look at the proof of Theorem I.14.7;
the bijection 9 given there is a homeomorphism.)
Notice that “cancellation properties” may not be true. For example,  ‚ Ö!× and  ‚ Ö!ß "× are
homeomorphic (both are countable discrete spaces) but topologically you can't “cancel the ” À Ö!× is
not homeomorphic to Ö0,1× !
Here are a few results which are quite simple but very handy to remember. The first states that
singleton factors are topologically irrelevant in a product.
Lemma 2.15
α−E \α ‚ " −F Ö:" × ¶ α−E \α
Proof " −F Ö:" × is itself a one-point space Ö:}, so we only need to prove that
α−E \α ‚ Ö:× ¶ α−E \α . The map 0 ÐBß :Ñ œ B is clearly a homeomorphismÞ ñ
Lemma 2.16 Suppose α−E \α Á g. For any F © E, α−F \α is homeomorphic to a subspace ^
of α−E \α  that is, α−F \α can be embedded in α−E \αÞ In fact, if all the \α 's are X" -spaces,
then α−F \α is homeomorphic to a closed subspace ^ of α−E \α .
Proof Pick a point : œ Ð:α Ñ − α−E \α . Then by Lemma 2.15,
α−F \α ¶ α−F \α ‚ α−EF Ö:α × œ ^ © α−E \α
Now suppose all the \α 's are X" . If C − Ðα−E \α Ñ  ^ , then for some # − E  Fß C# Á :#Þ Since
\# is a X" space, there is an open set Y# in \# that contains C# but not :# . Then C −  Y#  and
 Y#  ∩ Ðα−F \α ‚ α−EF Ö:α ×Ñ œ g.
Therefore ^ is closed in α−E \α . ñ
Note: 1) Assume all the \α 's œ ‘. Go through the preceding proof step-by-step when E œ Ö"ß ÞÞÞß 5×
and when E œ  and
2) In the case F œ Öα! ×, Lemma 2.16 says that each factor \α! is homeomorphic to subspace
of α−E \α (a closed subspace if all the \α 's are X" ).
3) But Lemma 2.16 does not say that if all the \α 's are X" , then every embedded copy of

\α! in α−E \α is closed: only that there exists a closed homeomorphic copy. (It is very easy to show
a copy of ‘ embedded in ‘# that is not closed in ‘# , for example ... ?)
Lemma 2.17 Suppose \ œ Ö\α À α − E× Á g. Then \ is a X" space (or, X# space) iff every
factor \α is X" (or, X# ).
246
Proof for Hausdorff Suppose all the \α 's are Hausdorff. If B Á C − \ , then Bα! Á Cα! for some
α! . Pick disjoint open sets Yα! and Zα! in \α! containing Bα! and Cα! .
Then

 Yα!  and  Zα!  are disjoint (basic) open sets in \α that contain B and C.
Conversely, suppose \ Á g. By Lemma 2.16, any factor \α is homeomorphic to a subspace of \ .
Since a subspace of a Hausdorff space is Hausdorff (why? Ñ, \α is Hausdorff. ñ
Exercise 2.18 Prove Lemma 2.17 if “Hausdorff” is replaced by “X" .” (The proof is similar but
easier.)
Theorem 2.19 The product of countably many 2-point discrete spaces is homeomorphic to the Cantor
set G .
Proof We show that if \8 œ Ö!ß #×, then ∞
8œ" \8 ¶ G .
To construct G we defined, for each sequence ÐB" ß B# ß ÞÞÞß B8 ß ÞÞÞÑ − Ö!ß #× , a descending
sequence of closed sets JB" ª JB" B# ª ÞÞÞ ª JB" B# ÞÞÞB8 ª ÞÞÞ in Ò!ß "Ó whose intersection gave a unique
point : − G À Ö:× œ ∞
8œ" JB" B# ÞÞÞB8 (see Section IV.10). For each 8, we can write G as a union of
8
# disjoint clopen sets: G œ ÐB" ßÞÞÞßB8Ñ−Ö!ß#×8 ÐG ∩ JB" B# ÞÞÞB8 Ñ.
Define0 À G Ä ∞
8œ" \8 by 0 Ð:Ñ œ B œ ÐB" ß B# ß ÞÞÞß B8 ß ÞÞÞÑÞ Clearly 0 is one-to-one and
onto. To show that 0 is continuous at : − G , it is sufficient to show that for each 8, the function
18 ‰ 0 À G Ä Ö!ß #× is continuous at :, so pick any 8. One of the clopen sets G ∩ JB" B# ÞÞÞB8 contains :
(call it Y ) and Ð18 ‰ 0 ÑlY œ B8 . Thus, 18 ‰ 0 is constant on a neighborhood of : in G , so 18 ‰ 0 is
continuous at :.
By Lemma 2.17, ∞
8œ" \8 is Hausdorff. Since 0 is a continuous bijection of a compact space
onto a Hausdorff space, 0 is a homeomorphism (why? ). ñ
Corollary 2.20 Ö!ß #×i! is compact.
This corollary is a very special case of the Tychonoff Product Theorem which states that any product
of compact spaces is compact. The Tychonoff Product Theorem is much harder and will be proved in
Chapter IX.
Corollary 2.21 The Cantor set is homeomorphic to a product of countably many copies of itself.
Proof
By Exercise 2.14 above, G i! ¶ ÐÖ!ß #×i! Ñi! ¶ Ö!ß #×i! †i! ¶ Ö!ß #×i! ¶ GÞ The case of
8
G ¶ G for 8 −  is similar. ñ
Example 2.22 Convince yourself that each assertion is true:
1) If \ is the Sorgenfrey line, then \ ‚ \ is the Sorgenfrey plane (see Examples
III.5.3 and III.5.4).
2) Let W " be the unit circle in ‘# . Then W " ‚ Ò!ß "Ó is homeomorphic to the
cylinder ÖÐBß Cß DÑ − ‘$ À B#  C# œ "ß D − Ò!ß "Ó×Þ
3) W " ‚ W " is homeomorphic to a torus ( œ “the surface of a doughnut”).
247
Exercises
E1. Does it ever happen that \ ‚ Ö!× open in \ ‚ ‘? If so, what is a necessary and sufficient
condition on \ for this to happen?
E2. a) Suppose \ and ] are topological spaces and that E © \ß F © ] Þ Prove that
int\‚] (E ‚ FÑ œ int\ E ‚ int] F: that is, “the interior of the product is the product of the interiors.”
(By induction, the same result holds for any finite product.) Give an example to show that the
statement may be false for infinite products.
b) Suppose Eα © \α for all α − E. Prove that in the product \ œ \α ,
cl (Eα Ñ œ  cl Eα .
Note: When the Eα 's are closed, this shows that Eα is closed: so “any product of closed sets is
closed.” Can you see any plausible reason why products of closures are better behaved than products
of interiors?
c) Suppose \ œ \α Á g and that Eα © \α . Prove that Eα is dense in \ iff Eα is dense in
\α for each α. Note: Part c) implies that a finite product of separable spaces is separable.
Part c) doesn't tell us whether or not an infinite product of separable spaces is separable: why not?
d) For each α, let ;α − \α . Prove that F œ ÖB − \α À Bα œ ;α for all but at most finitely many
α× is dense in \α . (Note: Suppose \ œ ‘ œ 8− \8 where each \8 œ ‘Þ Suppose each ;8 is
chosen to be a rational  say ;8 œ !. What does d) say about ‘ ?)
e) Let α! − E. Prove that ]α! œ ÖB − \α À Bα œ :α for all α Á α! × is homeomorphic
to \α! . Note: So any factor of a product has a “copy” of itself inside the product in a “natural” way.
For example, in ‘8 , the set of points where all coordinates except the first are ! is homeomorphic to
the first factor, ‘.Þ
f) Give an example of infinite spaces \ß ] ß ^ such that \ ‚ ] is homeomorphic to \ ‚ ^ but ]
is not homeomorphic to ^ .
E3. Let \ be a topological space and consider the “diagonal” set
? œ ÖÐBß BÑ À B − \× © \ ‚ \Þ
a) Prove that ? is closed in \ ‚ \ iff \ is Hausdorff.
b) Prove that ? is open in \ ‚ \ iff \ is discrete.
E4. Suppose \ is a Hausdorff space and that \α © \ for each α − E.. Show that
] œ Ö\α À α − E× is homeomorphic to a closed subspace of the product Ö\α À α − E×.
248
E5. For each 8 − , suppose H8 is a countable dense set in Ð\8 ß g8 ÑÞ
a) Prove that H œ H8 is dense in \8 .
b) Prove that \8 is separable. (Caution: H may not be countable (when is H countable?
Hint: In a product “closeness depends on only finitely many coordinates.” )
.
E6.
a) Suppose a3ß4 − Ö!ß #× for all 3ß 4 − . Prove that there exists a sequence Ðn5 Ñ in  such that,
for each 3, lim a3,85 exists.
5Ä∞
Hint: “picture” the +3ß4 in an infinite matrix. For each fixed 4, the “j-th column" of the matrix is a
point in the Cantor set G œ Ö!ß #×i! , and G is a compact metric space.
b) Let  +8 be an absolutely convergent sequence of real numbers. A series  +8w where each
∞
∞
+8 œ +8 or +8w œ ! is called a subseries of  +8 .
8œ"
8œ"
∞
w
Prove that W œ Ö= − ‘ À = is the sum of a subseries of  +8 × is closed in ‘.
8œ"
∞
Hint: “Absolute convergence” just guarantees that every subseries converges. Each subseries  +8w
8œ"
∞
8œ"
can be associated in a natural way with a point B − Ö!ß "×i! Þ Consider the mapping 0 À Ö!ß "×i! Ä ‘
given by 0 ÐBÑ œ  +8w − ‘Þ Must 0 be a homeomorphism?
∞
8œ"
c) Suppose K is an open dense subset of the Cantor set G . Must Fr K be countable?
Hint: Consider ÖÐ+ß ,Ñ − G ‚ G À , Á !×Þ
E7. Let  have the cofinite topology.
a) Does the product 7 have the cofinite topology?
Does the answer depend on 7?
b) Prove 7 is separable (Hint: When 7 is infinite, consider the simplest possible points in
the product.Ñ
Part b) implies that an arbitrarily large product of X" spaces with more than one point can
be separable. According to Theorem 3.8, this statement is false for X# spaces.
249
E8. In any set \ , we can define a topology by choosing a nonempty family of subsets Y and defining
closed sets to be all sets which can be written as an intersection of finite unions of sets from Y .
Y is called a subbase for the closed sets of \ . (This construction is “complementary” to generating a
topology on \ by using a collection of sets as a subbase for the open sets.)
a) Verify that this procedure does give a topology on \ .
b) Suppose \α is a topological space. Give \α the topology for which collection of “closed
boxes” Y œ ÖJα À Jα closed in \α × is a subbase for the closed sets. Is this topology the product
topology?
E9. Prove or disprove:
There exists a bijection 0 À \ œ Ö!ß "×i! Ä i! œ ] such that for all B − \ and for all 8,
the first 8 coordinates of C œ 0 ÐBÑ are determined by the first 7 coordinates of B.
Here, 7 depends on 8 and B. More formally, we are asking whether there exists a bijection
0 such that:
aB − \ a8 −  b7 −  such that changing B4 for any 4  7 does not change C5
for 5 Ÿ 8
(Hint: Think about continuity and the definition of the product topology.)
250
3. Productive Properties
We want to consider how some familiar topological properties behave with respect to products.
Definition 3.1 Suppose that, for each α − E, the space \α has a certain property T . We say that
 productive
if \α must have property T
if \α must have property T when E is countable
countably
productive
the property T is 
 finitely productive
if \α must have property T when E is finite
For example, Lemma 2.17 shows that the X" and X# properties are productive.
Some topological properties behave very badly with respect to products. For example, the Lindelof
¨
property is a “countability property” of spaces, and we might expect the Lindelof
¨ property to be
countably productive. Unfortunately, this is not the case.
Example 3.2 The Lindelof
¨ property is not finitely productive; in fact if \ is a Lindelof
¨ space, then
\ ‚ \ may not be Lindelof.
Let \ be the set of real numbers with the topology for which a
¨
neighborhood base at + is U+ œ ÖÒ+ß ,Ñ À +ß , − W , ,  +×. (Recall that \ is called the Sorgenfrey
Line: see Example III.5.3.) We begin by showing that f is Lindelof.
¨
It is sufficient to show that a collection h of basic open sets covering \ has a countable
subcover. Given such a cover h , Let i œ ÖÐ+ß ,Ñ À Ò+ß ,Ñ − h × and define E œ i . For a
moment, we can think of E as a subspace of ‘ with its usual topology. Then E is Lindelof
¨
(why?), and i is a covering of E by usual open sets, so there is a countable subfamily i w with
i w œ E.
Replace the left endpoints of the intervals in i w to get h w œ ÖÒ+ß ,Ñ À Ð+ß ,Ñ − i w × © h . If h w
covers \ we are done, so suppose \  h w Á g. For each B − \  h w , pick a set Ò+ß ,Ñ in
h that contains B. In fact, B must be the left endpoint of Ò+ß ,Ñ  for if B − Ò+ß ,Ñ − h and
B Á +, then B − i œ i w © h w Þ So, for each B  h w we can pick a set [B, ,B ) − h .
If B and C are distinct points not in h w , then ÒB, ,B ) and ÒC, ,C ) must be disjoint (why?). But
there can be at most countably many disjoint intervals ÒBß ,B ÑÞ So h w ∪ ÖÒB, ,B ) À B Â h w } is
a countable subcollection of h that covers W .
However the Sorgenfrey plane W ‚ W is not Lindelof.
¨ If it were, then its closed subspace
H œ ÖÐBß CÑ À B  C œ "× would also be Lindelof
¨ (Theorem III.7.10). But that is impossible
since H is uncountable and discrete in the subspace topology. (See the figure on the following
page.)
251
Fortunately, many other topological properties interact more nicely with products. Here are several
topological properties T to which the “same” theorem applies. We combine these into one large
theorem for efficiency.
Theorem 3.3 Suppose \ œ Ö\α À α − E× Á g. Let T be any one of the properties “first
countable,” “second countable,” “metrizable,” or “completely metrizable.” Then \ has property T iff
1) all the \α 's have property T , and
2) there are only countably \α 's that do not have the trivial topology.
This theorem “essentially” is a statement about countable products because:
1) The nontrivial \α 's are the “interesting” factors, and 2) says there are only countably
many of them. In practice, one hardly ever works with trivial spaces; and if we completely exclude
trivial spaces from the discussion, then the theorem just states that \ has property T iff \ is a
countable product of spaces with property T .
2) A nonempty X" space \α has the trivial topology iff l\α l œ "Þ So, if we deal
only with X" spaces (as is most often the case) the theorem says that \ has property T iff all the \α 's
have property T and all but countably many of the \α 's are “topologically irrelevant” singletons.
Of course, in the case of metrizability (or complete metrizability), the X" condition is automatically
satisfied.
Proof Throughout, let F œ the set of “interesting” indices œ Öα − E À \α is a nontrivial space×.
We begin with the case T œ “first countable.”
Suppose 1) and 2) hold and B − \. We need to produce a countable neighborhood base at B.
For each α − F , let ÖYα8 À 8 − × be a countable open neighborhood base at Bα − \α Þ Let
UB œ ÖY À Y is a finite intersection of sets of the form  Yα8  ß α − Fß 8 − ×
Since F is countable, UB is a countable collection of open sets containing B and we claim that
UB is a neighborhood base at B − \. To see this, suppose Z œ  Zα" ß ÞÞÞß Zα5  is a basic
open set containing B. (We may assume that all α3 's are in F À why?). For each 3 œ "ß ÞÞÞß 5ß
8
pick Y α833 so that Bα3 − Y α833 © Zα3 Þ Then Y œ  Yα8"" ß Yα8## ß ÞÞÞß Yαk5  − UB and B − Y © Z Þ
252
Conversely, suppose \ is first countable. We need to prove that 1) and 2) hold.
Since \ Á g, Lemma 2.16 gives that each \α is homeomorphic to a subspace of \ . Therefore
each \α is first countable, so 1) holds.
We prove the contrapositive for 2): assuming F is uncountable, we find a point B − \ at
which there cannot be a countable neighborhood base. Pick a point : œ Ð:α Ñ − \
For each " − Fß pick an open set S" © \" for which g Á S" Á \" .
B" − S" and C"  S" Þ Define B œ ÐBα Ñ − \ using the coordinates
B
Bα œ  "
:α − \α
Choose
if α œ " − F
if α Â F
Suppose UB is a countable collection of neighborhoods of B. For each R − UB choose a basic
open set Y with B − Y œ  Yα" ß ÞÞÞß Yα5  © R . There are only finitely many α3 's involved
in the expression for the chosen Y , and there are only countably many R 's in UB . So, since F
is uncountable, we can pick a " − F that is not one of the α3 's involved in the expression for
any of the chosen sets Y . Then
i)  S"  is an open set that contains B because B" − S"
ii) for all R − UB ß R ©
Î  S"  : to see this, consider the point A œ ÐAα Ñ
which is the same as B except in the " coordinate:
Aα œ 
Bα
C"
if α Á "
if α œ "
For each R − FB , we chose Y œ  Yα" ß ÞÞÞß Yα5  © R . Since B − Y and A has the same
α" ,..., α8 coordinates as B, we have A − Y also. But A   S"  because A" œ C"  S" Þ
Since Y ©
Î  S"  , certainly R ©
Î  S"  Þ So UB is not a neighborhood base at B.
We are now able to see immediately that if the product \ œ \α has any of the other properties T ,
then conditions 1) and 2) must hold.
If \ is second countable, metrizable or completely metrizable, then \ is first countable and,
by the first part of the proof, condition 2) must hold.
If \ is second countable or metrizable then every subspace has these same properties  in
particular each \α is second countable or metrizable respectively. If \ is completely
metrizable, then \ and all the subspaces \α are X" . By Lemma 2Þ16, \α is homeomorphic to
a closed  therefore complete  subspace of \ . Therefore \α is completely metrizable.
It remains to show that if T œ “second countable,” “metrizable,” or “completely metrizable” and
conditions 1) and 2) hold, then \ œ \α also has property T .
253
Suppose T œ “second countable.”
For each α in the countable set F , let Uα œ ÖSα" ß Sα# ß ÞÞÞß Sα8 ß ÞÞÞ } be a countable base for the
open sets in \α and let
U œ ÖS À S is a finite intersection of sets of form  Sα8  , where α − F and 8 − }
U is countable and we claim U is a base for the product topology on \ .
Suppose B − Z œ  Zα" ß ÞÞÞß Zα5  , a basic open set in \ . For each 3 œ "ß ÞÞÞß 5ß
Bα3 − Zα3 so we can choose a basic open set in \α3 such that Bα3 − Sα833 © Zα3 . Then
B −  Sα8"" ß Sα8## ß ÞÞÞß Sα855  © Z and  Sα8"" ß Sα8## ß ÞÞÞß Sα855  − U . Therefore Z can be
written as a union of sets from U , so U is a base for \ .
Suppose T œ “metrizableÞ”
Since all the \α 's are X" , condition 2) implies that all but countably many \α 's are singletons
and dropping the singleton factors from the product does not change \ topologically.
Therefore it is sufficient to prove that if each space \" ß \# ß ÞÞÞß \8 ß ÞÞÞ is metrizable, then
\ œ ∞
8œ" \8 is metrizable.
Let .8 be a metric for \8 ß where without loss of generality, we can assume each .8 Ÿ "
(why?). For points B œ ÐB8 Ñ, C œ ÐC8 Ñ − \ , define
.ÐBß CÑ œ 
∞
8œ"
.8 ÐB8 ßC8 Ñ
#8
Then . is a metric on \ (check ), and we claim that g. is the product topology g . Because \
is a countable product of first countable spaces, g is first countable, so g can be described
using sequences: so it is sufficient to show that ÐD5 Ñ Ä D in Ð\ß g Ñ iff ÐD5 Ñ Ä D in Ð\ß g. ÑÞ
But ÐD5 Ñ Ä D in Ð\ß g Ñ iff the ÐD5 Ñ converges “coordinatewise.” Therefore it is sufficient to
show that:
(D5 Ñ Ä D in Ð\ß g. Ñ
iff
a8 ÐD5 Ð8ÑÑ Ä DÐ8Ñ in Ð\8 ß .8 Ñ, that is,
a8 .8 ÐD5 Ð8Ñß DÐ8ÑÑ Ä !
i) Suppose .ÐD5 ß DÑ Ä !. Let %  ! and consider a particular 8! . We can choose O
so that 5 O implies .ÐD5 ß DÑ œ 
∞
8œ"
.8 ÐD5 Ð8Ñß DÐ8ÑÑ
#8

%
#8! .
Therefore, if 5 O
.8 ÐD5 Ð8! Ñß DÐ8! ÑÑ
certainly ! #8!
 #%8! , and therefore so .8! ÐD5 Ð8! Ñß DÐ8! ÑÑ  %.
So .8! ÐD5 Ð8! Ñß DÐ8! ÑÑ Ä !.
ii) On the other hand, suppose .8 ÐD5 Ð8Ñß DÐ8ÑÑ Ä ! for every 8. Let %  !. Choose
R so that 
∞
8œR "
"
#8

%
#
and then choose O so that if 5 O
%
." ÐD5 Ð"Ñß DÐ"ÑÑÎ#"  #R
%
.# ÐD5 Ð#Ñß DÐ#ÑÑÎ##  #R
ÞÞÞ
.R ÐD5 ÐR Ñß DÐR ÑÑÎ#R 
Then for 5 O we have .ÐD5 ß DÑ œ 
∞
8œ"
254
%
#R
.8 ÐD5 Ð8Ñß DÐ8ÑÑ
#8
.
œ
R
8œ"
.8 ÐD5 Ð8Ñß DÐ8ÑÑ
#8

∞
8œR "
.8 ÐD5 Ð8Ñß DÐ8ÑÑ
#8
R†
%
#R

%
#
œ %Þ
Therefore .ÐD5 ß DÑ Ä !.
Suppose T œ “completely metrizableÞ”
Just as for T œ “metrizable,” we can assume \ œ ∞
8œ" \8 and that .8 is a complete metric
on \8 with .8 Ÿ ". Using these .8 's, we define . in the same way. Then g. is the product
topology on \ . We only need to show that Ð\ß .Ñ is complete.
Suppose ÐD5 Ñ is a Cauchy sequence in Ð\ß .Ñ. From the definition of . it is easy to see that
ÐD5 Ð8ÑÑ is Cauchy in Ð\8 ß .8 Ñ for each 8, so that ÐD5 Ð8ÑÑ Ä some point +8 − \8 Þ
Let + œ Ð+8 Ñ − \ . Since ÐD5 Ð8ÑÑ Ä +8 for each 8, we have ÐD5 Ñ Ä + in the product topology
œ g. . Therefore Ð\ß .Ñ is complete. ñ
What is the correct formulation and proof of the theorem for T œ “pseudometrizable” ?
We might wonder why T œ “separable” is not included with Theorem 3.3. Since separability is a
“countability property,” we might hope that separability is preserved in countable products  although
our experience Lindelof
¨ spaces could make us hesitate. The reason for the omission is that separability
is better behaved for products than the other properties. (You should try to prove directly that a
countable product of separable spaces is separable  remembering that in the product topology,
“closeness depends on finitely many coordinates.” If necessary, first look at finite products.) But
surprisingly, the product of as many as - separable spaces must be separable, and the product of more
than - nontrivial separable spaces can sometimes be separable.
We begin the treatment of separability and products with a simple lemma which is really nothing but
set theory.
Lemma 3.4 Suppose lEl Ÿ -Þ There exists a countable collection e of subsets of E with the
following property: given distinct α" , α# , ... , α8 − E, there are disjoint sets E" ß E# ß ÞÞÞß E8 − e such
that α3 − E3 for each 3.
Proof (Think about how you would prove the theorem when E œ ‘Þ The general case is just a
“carry over” on the same idea.)
Since lEl Ÿ - , there is a one-to-one map 9 À E Ä ‘. Let e œ Ö9" Ò Ð+ß ,Ñ Ó À +ß , − ×Þ
For distinct α" , α# ,..., α8 − E, 9Ðα" Ñ, 9Ðα# Ñ, ..., 9Ðα8 Ñ are distinct real numbers; we can choose
+3 , ,3 −  so that 9Ðα3 Ñ − Ð+3 ß ,3 Ñ and so that the intervals Ð+3 ß ,3 Ñ are pairwise disjoint. Then the sets
E3 œ 9" ÒÐ+3 ß ,3 ÑÓ − e are the ones we need. ñ
Theorem 3.5
1) Suppose \ œ α−E \α Á g. If \ is separable, then each \α is separable.
2) If each \α is separable and lEl Ÿ - , then \ œ α−E \α is separable.
Part 2) of Theorem 3.5 is attributed (independently) to several people. In a slightly more general
version, it is sometimes called the “Hewitt-Marczewski-Pondiczery Theorem.” Here is an amusing
255
sidelight, written by topologist Melvin Henriksen online in the Topology Atlas. A few words have
been modified to conform with our notation:
Most topologists are familiar with the Hewitt-Marczewski-Pondiczery theorem. It states that if
7 is an infinite cardinal, then a product of #7 topological spaces, each of which has a dense
set of cardinality Ÿ 7ß also has a dense set with Ÿ 7 points. In particular, the product of separable spaces is separable (where - is the cardinal number of the continuum). Hewitt's
proof appeared in [Bull. Amer. Math. Soc. 52 (1946), 641-643], Marczewski's proof in [Fund.
Math. 34 (1947), 127-143], and Pondiczery's in [Duke Math. 11 (1944), 835-837]. A proof
and a few historical remarks appear in Chapter 2 of Engelking's General Topology. The
spread in the publication dates is due to dislocations caused by the Second World War; there
is no doubt that these discoveries were made independently.
Hewitt and Marczewski are well-known as contributors to general topology, but who was (or
is) Pondiczery? The answer may be found in Lion Hunting & Other Mathematical Pursuitsß
edited by G. Alexanderson and D. Mugler, Mathematical Association of America, 1995. It is a
collection of memorabilia about Ralph P. Boas Jr. (1912-1992), whose accomplishments
included writing many papers in mathematical analysis as well as several books, making a lot
of expository contributions to the American Mathematical Monthly, being an accomplished
administrator (e.g., he was the first editor of Mathematical Reviews (MR) who set the tone for
this vitally important publication, and was the chairman for the Mathematics Department at
Northwestern University for many years and helped to improve its already high quality), and
helping us all to see that there is a lot of humor in what we do. He wrote many humorous
articles under pseudonyms, sometimes jointly with others. The most famous is “A Contribution
to the Mathematical Theory of Big Game Hunting” by H. Petard that appeared in the Monthly
in 1938. This book is a delight to read.
In this book, Ralph Boas confesses that he concocted the name from Pondicheree (a place in
India fought over by the Dutch, English and French), changed the spelling to make it sound
Slavic, and added the initials E.S. because he contemplated writing spoofs on extra-sensory
perception under the name E.S. Pondiczery. Instead, Pondiczery wrote notes in the Monthly,
reviews for MR, and the paper that is the subject of this article. It is the only one reviewed in
MR credited to this pseudonymous author.
One mystery remains. Did Ralph Boas have a collaborator in writing this paper? He certainly
had the talent to write it himself, but facts cannot be established by deduction alone. His son
Harold (also a mathematician) does not know the answer to this question...
Proof 1) Let H be a countable dense set in \ . For each αß 1α ÒHÓ is countable and dense in \ α
(because \α œ 1α Ò\Ó œ 1α Òcl HÓ © cl 1α ÒHÓ ). Therefore each \α is separable.
2) Choose a family e as in Lemma 3.4 and for each α, let H α œ ÖB1α , B2α ß ÞÞÞß B8α ß ÞÞÞ × be a
countable dense set in \α . Define a countable set f by
f œ ÖÐE" ß ÞÞÞß E8 ß 6" ß ÞÞÞß 68 Ñ À 8 − ß 63 − , E3 − e with the E3 's pairwise disjoint×Þ
Pick a point :α − \α for each α. For each #8-tuple = œ ÐE" ß ÞÞÞß E8 ß 6" ß ÞÞÞß 68 Ñ − f , define a point
B= − \ with coordinates
256
B= ÐαÑ œ 
Bα63
:α
if α − E3
if α Â E3
Let H œ ÖB= À = − f ×. H is countable and we claim that H is dense in \ . To see this, consider any
nonempty basic open set Y œ  Yα" ß ÞÞÞß Yα5  À we will show that Y ∩ H Á gÞ
For Y œ  Yα" ß ÞÞÞß Yα5  Á g,
i) Choose disjoint sets E" ß ÞÞÞß E5 in e so that α" − E" ß ÞÞÞß α5 − E5
ii) For each 3 œ "ß ÞÞÞß 5ß Hα3 is dense in \α3 and we can pick a point Bα833 in Hα3 ∩ Yα3
œ ÖB"α3 , B#α3 ß ÞÞÞß B8α3 ß ÞÞÞ × ∩ Yα3 Þ
Let = œ ÐE" ß ÞÞÞß E3 ß ÞÞÞß E5 ß 8" ß ÞÞÞ83 ß ÞÞÞß 85 Ñ − f Þ Because α3 − E3 , we have B= Ðα3 Ñ œ B8α33 − Yα3 .
Therefore B= − Y ∩ H. ñ
Example 3.6 The rather abstract construction of a dense set H in the proof of Theorem 3.5 can be
nicely illustrated with a concrete example. Consider ‘‘ Ð œ <−‘ \< , where each \< œ ‘ÑÞ Choose
e to be the collection of all open intervals Ð+ß ,Ñ with rational endpoints, and make a list these intervals
as E" ß ÞÞÞß E8 ß ÞÞÞ . In each \< ß choose H< œ  œ Ö; " ß ; # ß ÞÞÞ; 8 ß ÞÞÞ×Þ (Since all the H< 's are identical,
we can omit the subscript “<” on the points; but just to stay consistent with the notation i8 the proof,
we still index the ; 's using superscripts .) For each <, (arbitrarily) pick :< œ ! − \< Þ
One example of a 6-tuple in the collection f is = œ ÐE' ß E# ß E& ß #ß (ß %Ñ, where E" ß E# ß E$ are disjoint
open intervals with rational endpoints. The corresponding point B= − ‘‘ is the function B= À ‘ Ä ‘
with
#


 ;(
;
B= Ð<Ñ œ  %

;
!
for < − E'
for < − E#
for < − E&
for < Â E' ∪ E& ∪ E#
The dense set H consists of all step functions (such as B= ) that are ! outside a finite union
E8" ∪ E8# ∪ ÞÞÞ ∪ E85 of disjoint open intervals with rational endpoints and which have a constant
rational value on each E3 .
Caution: In Example 2.12 we saw that the product topology on ‘‘ is the topology of pointwise
convergence  that is, Ð08 Ñ Ä 0 in ‘‘ iff Ð08 Ð<ÑÑ Ä 0 Ð<Ñ for each < − ‘. But ‘‘ is not first
countable (why?) so we cannot say that sequences are sufficient to describe the topology. In particular,
if 0 − ‘‘ ß then 0 − cl H but we cannot say that there must be sequence of step functions from H that
converges pointwise to 0 Þ
Since ‘‘ is not first countable, ‘‘ is an example of a separable space that is neither second countable
nor metrizable.
In contrast to the properties discussed in Theorem 3.3, an arbitrarily large product of nontrivial
separable spaces can sometimes turn out to be separable, as the next example shows. However,
257
Theorem 3.8 shows that for Hausdorff spaces, a nonempty product with more than - factors cannot be
separable.
Example 3.7 For each α − E, let \α be a set with l\α l  "Þ Choose :α − \α and let
gα œ ÖS © \α À :α − S× ∪ Ög×Þ The singleton set Ö:α × is dense in Ð\α ß gα Ñ, so \α is separable.
If : œ Ð:α Ñ − \ œ α−E \α , then singleton set Ö:× is dense (why? look at a nonempty basic
open set Y .) So \ is separable, no matter how large the index set E is.
The \α 's in this example are not X" . But Exercise E7 shows that an arbitrarily large
product of separable X" spaces can turn out to be separable.
Theorem 3.8 Suppose \ œ α−E \α Á g where each \α is a X# space with more than one point.
If \ is separable, then lEl Ÿ -Þ
Proof For each α, we can pick a pair of disjoint, nonempty open sets Yα and Zα in \α Þ Let H be a
countable dense set in \ and let Hα œ  Yα  ∩ H for each α. If α Á " − Eß there is a point
: −  Yα ß Z"  ∩ H because H is dense. Then : − Hα but :  H" œ  Y"  ∩ H since
B"  Y" À therefore Hα Á H" . Therefore the map 9 À E Ä c ÐHÑ given by 9ÐαÑ œ Hα is one-toone, so lEl Ÿ lc ÐHÑl œ #i! œ -Þ ñ
We saw in Corollary V.2.19 that a finite product of connected spaces is connected. In fact, the
following theorem shows that connectedness behaves very nicely with respect to any product. The
proof of the theorem is interesting because, unlike our previous proofs about products, this proof uses
the theorem about finite products to prove the general case.
Theorem 3.9 Suppose \ œ α−E \α Á g. \ is connected iff each \α is connected.
Proof \ is nonempty so each map 1α À \ Ä \α is onto. If \ is connected, then each \α is a
continuous image of a connected space so \α is connected (see Example V.2.6).
Conversely, suppose each \α is connectedÞ For each α, pick a point :α − \α . For each finite
set J © E, let \J œ α−J \α ‚ α−EJ Ö:α ×. \J is homeomorphic to the finite product α−J \α ,
so each \J is connected. Let H œ Ö\J À J is a finite subset of E×. Each \J contains the point
: œ Ð:α Ñ, so Corollary V.2.10 tells us that H is connected. Unfortunately, H Á \ (except in trivial
cases; why?).
But we claim that H is dense in \ . If Y œ  Yα" ß ÞÞÞß Yα8  is any nonempty basic open set in \ ,
we need to show that H ∩ Y Á g. Choose a point Bα3 − Yα3 for each 3 œ "ß ÞÞÞß 8, and define B − \
having coordinates
B
if α œ α3
BÐαÑ œ  α3
:α if α Á α" , α# ,..., α8
If we let J œ Ö+" ß α# ß ÞÞÞß α8 ×, then B − \J ∩ Y © H ∩ Y , so H ∩ Y Á gÞ
Therefore \ œ cl H is connected (Corollary V.2.20). ñ
Question: is the analogue of Theorem 3.9 true for path connected spaces?
Just for reference, we state one more theorem here. We will not prove Theorem 3.10 until Chapter IX,
but we may use it in examples. (Of course, the proof in Chapter IX will not depend on any of these
258
examples! )
Theorem 3.10 (Tychonoff Product Theorem) Suppose \ œ α−E \α Á g. Then \ is compact iff
each \α is compact.
The proof “ Ê ” in Tychonoff's Theorem is quite easy (why?). And the easy theorem that a finite
product of compact spaces is compact was in Exercise IV.E.26. .
259
Exercises
E10. Let \ œ Ò!ß "ÓÒ!ß"Ó have the product topology.
a) Prove that the set of all functions in \ with finite range (sometimes called step functions) is
dense in \ .
b) By Theorem 3.5, \ is separable. Describe a countable set of step functions which is dense
in \ .
c) Let E œ ÖB − \ À 0 is the characteristic function of a singleton set Ö<× ×Þ
Prove that E, with the subspace topology, is discrete and not separable. Is E closed?
d) Prove that E has exactly one limit point, D , in \ and that if R is a neighborhood of D ,
then E  R is finite.
E11. “Boxes” of the form ÖYα À α − E×, where Yα is open in \α , are a base for the box topology
on α−E \α Þ {Ö\α À α − E×. Throughout this problem, we assume that products have the box
topology rather than the usual product topology.
a) Show that the “diagonal map” 0 À ‘ Ä ‘i! given by 0 ÐBÑ œ ÐBß Bß Bß ÞÞÞÑ is not continuous,
but that its composition with each projection map is continuous.
b) Show that Ò !ß "Ói! is not compact.
Hint: let E! œ Ò !ß "Ñ and E" œ Ð!ß "Ó. Consider the collection h of all sets of the form
E%" ‚ E%# ‚ ... ‚ E%8 ‚ ÞÞÞ , where Ð%" ß %# ß ÞÞÞß %8 ß ÞÞÞÑ − Ö!ß "×i! .
Note: [ ith the product topology, in contrast, Ò !ß "Ó7 is compact for any cardinal 7  by the
Tychonoff Product Theorem (3.10) which we will prove later.
c) Show that ‘i! is not connected by showing that the set E œ ÖB − ‘i! À B is an unbounded
sequence in ‘× is clopen.
d) Suppose Ð\ß .Ñ and Ð\α ß .α Ñ Ðα − EÑ are metric spaces. Prove that a function
0 À \ Ä \α (with the box topology) is continuous iff each coordinate function 0α œ 1α ‰ 0 is
continuous and each B − \ has a neighborhood on which all but a finite number of the 0α 's are
constant.
E12. State and prove a theorem that gives a necessary and sufficient condition for a product of spaces
to be path connected.
E13. Prove the following more general version of Theorem 3.8:
Suppose \ œ α−E \α Á g, and that, for each α − E, there exist disjoint nonempty open sets
Yα and Zα in \α . If \ is separable, then lEl Ÿ -Þ
260
4. Embedding Spaces in Products
It is often possible to embed a space \ in a product \α Þ Such embeddings will give us some nice
theorems  for example, we will see that there is a separable metric space \ that contains all other
separable metric spaces topologically  a “universal” separable metric space.
To illustrate the general method we will use, consider two maps 0" À Ò!ß "Ó Ä ‘ and 0# À Ò!ß "Ó Ä ‘
given by 0" ÐBÑ œ B# and 0# ÐBÑ œ /B Þ Using these maps we can define / À Ò!ß "Ó Ä ‘ ‚ ‘ œ ‘# by
using 0" and 0# as “coordinate functions”: /ÐBÑ œ Ð0" ÐBÑß 0# ÐBÑÑ œ ÐB# ß /B ÑÞ This map / is called the
evaluation map defined by the set of functions Ö0" , 0# ×. In this example, / is an embedding  that is,
/ is a homeomorphism of Ò!ß "Ó into ‘# so that Ò!ß "Ó ¶ ranÐ/Ñ © ‘# (see the figure).
An evaluation map does not always give an embedding: for example, the evaluation map
/ À Ò!ß "Ó Ä ‘# defined by the family Öcos #1Bß sin #1B× is not a homeomorphism between Ò!ß "Ó and
ranÐ/Ñ © ‘# (why? what is ranÐ/Ñ? )
We want to generalize the idea of an evaluation map / into a product and to find some conditions under
which / will be an embedding.
Definition 4.1 Suppose \ and \α (α − EÑ are topological spaces and that 0α À \ Ä \α for each αÞ
The evaluation map defined by the family Ö0α À α − E× is the function / À \ Ä α−E \α given by
/ÐBÑÐαÑ œ 0α ÐBÑ
Then /ÐBÑ is the point in the product \α whose αth coordinate function is 0α ÐBÑ:\. In more informal
coordinate notation, /ÐBÑ œ Ð0α ÐBÑÑÞ
Exercise 4.2 Suppose \ œ α−E \α Þ For each α there is a projection map 1α À \ Ä \α . What is
the evaluation map defined by the family Ö1α À α − E× ?
261
Definition 4.3 Suppose \ and \α (α − EÑ are topological spaces and that 0α À \ Ä \α Þ We say that
the family Ö0α À α − E× separates points if, for each pair of points B Á C − \ , there exists an α − E
for which 0α ÐBÑ Á 0α ÐCÑ.
Clearly, the evaluation map / À \ Ä α−E \α is one-to-one Í for all B Á C − \ß /ÐBÑ Á /ÐCÑ
Í for all B Á C − \ there is an α − E for which /ÐBÑÐαÑ œ 0α ÐBÑ Á 0α ÐCÑ œ /ÐCÑÐαÑ
Í the family Ö0α À α − E× separates points.
Theorem 4.4 Suppose \ has the weak topology generated by the maps 0α À \ Ä Ð\α ß gα Ñ and that
the family Ö0α À α − E× separates points. Then / is an embedding  that is, / a homeomorphism
between \ and /Ò\Ó © \α .
Proof Since Ö0α × separates points, / is one-to-one and / is continuous because each composition
1α ‰ / œ 0α is continuous.
/ preserves unions and (since / is one-to-one) intersections. So to check that / is an open map
from \ to /Ò\Ó, it is sufficient to show that / takes subbasic open sets in \ to open sets in /Ò\Ó.
Because \ has the weak topology, a subbasic open set has the form Y œ 0α" ÒZ Ó, where Z is open in
\α . But then /ÒY Ó œ /Ò0α" ÒZ ÓÓ œ 1α" ÒZ Ó ∩ /Ò\Ó is an open set in /Ò\ÓÞ ñ
(Note: /ÒY Ó might not be open in \α , but that is irrelevant. See the earlier example where
/ÐBÑ œ ÐB# ß /B )Þ
The converse of Theorem 4.4 is also true: if e is an embedding, then the 0α 's separate points and \
has the weak topology generated by the 0α 's. However, we do not need this fact and will omit the proof
(which is not very hard).
Example 4.5
Let Ð\ß .Ñ be a separable metric space. We can assume, without loss of generality, that . Ÿ "Þ \ is
second countable so there is a countable base U œ ÖY" ß ÞÞÞß Y8 ß ÞÞÞ× for the open sets. For each 8, let
08 ÐBÑ œ .ÐBß \  Y8 ÑÞ Then 08 À \ Ä Ò!ß "Ó is continuous and, since \  Y8 is closed, we have
08 ÐBÑ  ! iff B − Y8 Þ If B Á C − \ , there is an 8 such that B − Y8 and C Â Y8 Þ Then
08 ÐCÑ œ ! Á 08 ÐBÑ, so 08 's separate points.
We claim that the topology g. on \ is actually just the weak topology gA generated by the 08 's.
Because the functions 08 are continuous if \ has the topology g. , we know that g. ª gA Þ To show
g. © gA , suppose B − Y − g. Þ For some 8, we have B − Y8 © Y and therefore 08 ÐBÑ œ -  !Þ But
Z8 œ 08" ÒÐ #- ß "ÓÓ is a subbasic open set in the weak topology and B − Z8 © Y8 © Y Þ Therefore
Y − gA .
By Theorem 4.4, / À \ Ä Ò!ß "Ói! is an embedding, so \ ¶ /Ò \ Ó © Ò!ß "Ói! Þ We sometimes write this
top
as \ © Ò!ß "Ói! . From Theorems 3.3 and 3.5, we know that Ò!ß "Ói! is itself a separable metrizable
space  and therefore all its subspaces are separable and metrizable. Putting all this together, we get
that topologically, the separable metrizable spaces are nothing more and nothing less than the
subspaces of Ò!ß "Ói! .
We can view this fact with a “half-full” or “half-empty” attitude:
262
i) separable metric spaces must not be very complicated since topologically they are nothing
more than the subspaces of a single very nice space: the “cube” Ò!ß "Ói!
ii) separable metric spaces can get quite complicated, so the subspaces of a cube Ò!ß "Ói!
are more complicated than we imagined.
"
Since Ò!ß "Ói! ¶ ∞
8œ" Ò!ß 8 Ó œ L (the “Hilbert cube”) © j# , we can also say that topologically
the separable metrizable spaces are precisely the subspaces of j# . This is particularly amusing
because of the almost “Euclidean” metric . on j# :
∞
"
.ÐBß CÑ œ Ð  ÐB8  C8 Ñ# Ñ #
8œ"
In some sense, this elegant “Euclidean-like” metric is adequate to describe the topology of any
separable metric space.
We can summarize by saying that each of L and j# is a “universal separable metric space” (But they
are not homeomorphic: L is compact but j# is not.)
Example 4.6
Suppose Ð\ß .Ñ is a metric space, not necessarily separable, and that ÖYα À α − E× is a base for the
topology g. , where 7 œ lElÞ An argument completely analogous to the reasoning in Example 4.5
top
(just replace “8” everywhere with “α”) shows that \ © Ò!ß "Ó7 . Therefore topologically every metric
space is a subspace of some sufficiently large “cube.” Of course, if 7  i! , the cube Ò!ß "Ó7 is not
itself metrizable (why? ); in general this cube will have many subspaces that are nonmetrizable. So the
result is not quite as dramatic as in the separable case.
The weight AÐ\Ñ of a topological space Ð\ß g Ñ is defined as minÖlU l À U is a base for g ×  i! Þ
We are assuming here that the “min” in the definition exists: see Example 5.22 in Chapter VIII.
J or some very simple spaces, the “min” could be finite  in which case the “  i! ” guarantees
that AÐ\Ñ i! Ðconvenient for purely technical reasons that don't matter in these notes).
The density $ Ð\Ñ is defined as minÖlHl À H is dense in Ð\ß g Ñ×  i! Þ For a metrizable space, it is
not hard to prove that AÐ\Ñ œ $ Ð\Ñ. The proof is just like our earlier proof (in Theorem III.6.5) that
separability and second countability are equivalent in metrizable spaces. Therefore we have that for
any metric space Ð\ß .Ñß
top
\ © Ò!ß "ÓAÐ\Ñ œ Ò!ß "Ó$ Ð\Ñ
(*)
Notice that, for a given space Ð\ß .Ñ, the exponents in this statement are not necessarily the smallest
top
possible. For example, (*) says that ‘ © Ò!ß "ÓAÐ‘Ñ œ Ò!ß "Ói! , but in fact we can do much better than
the exponent i! À ‘ ¶ Ð!ß "Ñ © Ò!ß "Ó œ Ò!ß "Ó" !
We add one additional comment, without proof: For a given infinite cardinal 7, it is possible to define
a metrizable space L7 with weight 7 such that every metric space \ with weight 7 can be
i!
embedded in Hi7! Þ In other words, L7
is a metrizable space which is “universal for all metric spaces
with AÐ\Ñ œ 7.” The price of metrizability, here, is that we need to replace Ò!ß "Ó by a more
complicated space L7 .
263
Without going into all the details, you can think of L7 as a “star” with 7 different copies of
Ò!ß "Ó (“rays”) all placed with ! at the center of the star. For two points Bß C on the same “ray”
of the star, .ÐBß CÑ œ lB  Cl; if Bß C are on different rays, the distance between them is
measured “via the origin” À .ÐBß CÑ œ lBl  lClÞ
The condition in Theorem 4.4 that “\ has the weak topology generated by a collection of maps
0α À \ Ä \α ” is sometimes not so easy to check. The following definition and theorem can
sometimes help.
Definition 4.7 Suppose \ and \α (α − EÑ are topological spaces and that, for each α, 0α À \ Ä \α Þ
We say that the collection Y œ Ö0α À α − E× separates points from closed sets if whenever J is a
closed set in \ and B Â J , there is an α such that 0α ÐBÑ Â cl 0α ÒJ ÓÞ
Example 4.8 Let \ œ ‘ and Y œ GБÑ. Suppose J is a closed set in ‘ and <  J . There is an
open interval Ð+ß ,Ñ for which < − Ð+ß ,Ñ © ‘  J . Define 0 − GÐ‘Ñ with a graph like the one shown
in the figure:
Then ! œ 0 Ð<Ñ Â cl 0 ÒJ ÓÞ Therefore GÐ‘Ñ separates points and closed sets.
The same notation continues in the following lemma.
Lemma 4.9 The family U œ Ö0α" ÒZ Ó À α − Eß Z open in \α × is a base for the topology in \ iff
i) the 0α 's are continuous, and
 ii) Ö0 À α − E× separates points and closed sets.
α
In particular, i)  ii) imply that U is a subbase for the topology on \  so that \ has the weak
topology generated by the 0α 's.
264
Note: the more open (closed) sets there are in \ , the harder it is for a given family Ö0α À α − E× to
succeed in separating points and closed sets. In factß the lemma shows that a family of continuous
functions Ö0α À α − E× succeeds in separating points and closed sets only if g is the smallest
topology that makes the 0α 's continuous.
Proof Suppose U is a base. Then the sets 0α" ÒZ Ó in U are open, so the 0α 's are continuous and
i) holds.
To prove ii), suppose J is closed in \ and B Â J . For some α and some Z open in \α , we
have B − 0α" ÒZ Ó © \  J Þ Then 0α ÐBÑ − Z and Z ∩ 0α ÒJ Ó œ g Ðsince 0α" ÒZ Ó ∩ J œ gÑ.
Therefore 0α ÐBÑ Â cl 0α ÒJ Ó so ii) also holds.
Conversely, suppose i) and ii) hold. If B − S and S is open in \ , we need to find a set
0α" ÒZ Ó − U such that B − 0α" ÒZ Ó © S Þ Since B  J œ \  S, condition ii) gives us an α for which
0α ÐBÑ Â cl 0α ÒJ Ó. Then 0α ÐBÑ − Z œ \α  cl 0α ÒJ ÓÞ Then B − 0α" ÒZ Ó and we claim 0α" ÒZ Ó © S:
Suppose D − 0α" ÒZ Ó so 0α ÐDÑ − Z œ \α  cl 0α ÒJ ÓÞ
But if D − J , then 0α ÐDÑ − 0α ÒJ Ó œ 0α Òcl J Ó © cl 0α ÒJ ÓÞ
Therefore D − \  J œ SÞ ñ
Theorem 4.10 Suppose 0α À \ Ä \α is continuous for each α − E. If the collection Ö0α À α − E×
i) separates points and closed sets, and
 ii) separates points
then the evaluation map / À \ Ä \α is an embedding.
Proof Since the 0α 's are continuous, Lemma 4.9 gives us that \ has the weak topology. Then
Theorem 4.4 implies that / is an embedding. ñ
If the space \ we are trying to embed is a X" -space (as is most often the case), then
i) Ê ii) in Theorem 4.10 , so we have the simpler statement given in the following corollary.
Corollary 4.11 Suppose 0α À \ Ä \α is continuous for each α − E. If \ is a X" -space and
Ö0α À α − E× separates points and closed sets, then evaluation map / À \ Ä \α is an embedding.
Proof By Theorem 4.10, it is sufficient to show that the 0α 's separate points, so suppose B Á C − \ .
Since B Â the closed set ÖC×, there is an α for which 0α ÐBÑ Â cl 0α ÒÖC×Ó. Therefore 0α ÐBÑ Á 0α ÐCÑß so
Ö0α À α − E× separates points. ñ
265
Exercises
E14. A space Ð\ß g Ñ is called a X! space if whenever B Á C − \ , then aB Á aC (equivalently, either
there is an open set Y containing B but not C, or vice-versa). Notice that the X! condition is weaker
than X" (see example III.2.6.4). Clearly, a subspace of a X! -space is X! .
a) Prove that a nonempty product \ œ Ö\α À α − E× is X! iff each \α is X! .
b) Let W be “Sierpinski space”  that is, W œ Ö!ß "× with the topology g œ Ögß Ö1×ß Ö!ß "××. Use
the embedding theorems to prove that a space \ is X! iff \ is homeomorphic to a subspace of W 7 for
some cardinal 7. (Nearly all interesting spaces are X! , so nearly all interesting spaces are
(topologically) just subspaces of W 7 for some cardinal 7.)
Hint Ê : for each open set Y in \ , let ;Y be the characteristic function of Y Þ Use an embedding
theorem.
E15. a) Let Ð\ß .Ñ be a metric space. Prove that GÐ\Ñ ( œ the collection of continuous real-valued
functions on \ ) separates points from closed setsÞ (Since \ is X" , GÐ\Ñ therefore also separates
points).
b) Suppose \ is any X! topological space for which GÐ\Ñ separates points and closed sets.
Prove that \ can be embedded in a product of copies of ‘.
E16. A space \ satisfies the countable chain condition (CCC) if every collection of nonempty
pairwise disjoint open sets must be countable. (For example, every separable space satisfies CCC.)
Suppose that \α is separable for each α − E . Prove that \ œ Ö\α : α − E× satisfies CCC.
(There isn't much to prove when | A | Ÿ c; why?)
Hint: Let ÖU> À > − X × be any such collection. We can assume all the U> 's are basic open sets (why?).
Prove that if W © T and | W | Ÿ c, then | W | Ÿ i! and hence X must be countable.)
E17. There are several ways to define dimension for topological spaces. One classical method is the
following inductive definition.
Define dim g œ  1.
For : − \ , we say that \ has dimension Ÿ 0 at p if there is a neighborhood base at p
consisting of sets with  1 dimensional (that is, empty) frontiers. Since a set has empty frontier iff
it is clopen, \ has dimension Ÿ 0 at : iff : has a neighborhood base consisting of clopen sets.
We say \ has dimension Ÿ 8 at : if : has a neighborhood base consisting of sets with n  1
dimensional frontiers.
We say \ has dimension Ÿ n, and write dimÐ\Ñ Ÿ n, if \ has dimension Ÿ 8 at p for each
p − X and that dim(X) œ n if dim(\ ) Ÿ 8 but dimÐ\Ñ Ÿ
Î 8  ". We say dimÐ\Ñ œ ∞ if
dimÐ\Ñ Ÿ 8 is false for every 8 − Þ
While this definition of dimÐ\Ñ makes sense for any topological space \ , it turns out that
“dim” produces a nicely behaved dimension theory only for separable metric spaces. The
dimension function “dim” is sometimes called small inductive dimension to distinguish it from
other more general definitions of dimension. The classic discussion of small inductive dimension
is in Dimension Theory (Hurewicz and Wallman).
266
It is clear that “dimÐ\Ñ œ 8” is a topological property of \ . There is a theorem stating that
dim( ‘8 ) œ 8, from which it follows that ‘8 is not homeomorphic to ‘7 if m Á n. In proving the
theorem, showing dim(Б8 Ñ Ÿ n is easy; the hard part is showing that dimБ8 Ñ Ÿ
Î n  1.
a) Prove that dim(‘) œ ".
b) Let G be the Cantor set. Prove that dim(G ) œ !Þ
c) Suppose Ð\ß .Ñ is a !-dimensional separable metric space. Prove that \ is homeomorphic
to a subspace of C. (Hint: Show that \ has a countable base of clopen sets. View G as Ö!ß #×i!
and apply the embedding theorems.)
E18. Suppose \ and ] are X! -spaces. Part a) outlines a sufficient condition enabling ] to be
top
embedded in a product of \ 's, i.e., that ] © \ 7 for some cardinal m. Parts b) and c) look at some
corollaries.
a) Theorem Let \ and ] be X! spaces. Then ] can be topologically embedded in \ 7 for
some cardinal 7 if, for every closed set J © ] and every point C Â J , there exists a
continuous function 0 À ] Ä \ 8 such that f (y) Â cl f [F] (here, n is finite and depends on f ).
Proof Let X œ Ö> œ ÐCß J Ñ À J is closed in ] and C  J × and, for each such pair
> œ ÐCß 0 Ñ, let 0> be the function given in the hypothesis. Let \> œ \ 8 (the space
containing the values of 0> ). Then clearly Ö\> À > − X × is homeomorphic to \ 7 for some
7, so it suffices to show ] can be embedded in Ö\> À > − X ×.
Let
2 À ] Ä Ö\> À > − X × be defined as follows: for C − ] , 2Ð>Ñ has for its >-th coordinate
0> ÐCÑß i.e., 2ÐCÑÐ>Ñ œ 0> ÐCÑÞ
i) Show 2 is continuous.
ii) Show 2 is one-to-one.
iii) Show that 2 is a closed mapping onto its range 2Ò] Ó to complete the proof that 2 is
+ homeomorphism between ] and 2Ò] Ó © Ö\> À > − X ×. (Note: the converse of the
theorem is also true. Both the theorem and converse are due to S. Mrowka.)
b) Let F denote Sierpinski space {{0,1},{g,{0},{0,1}}. Use the theorem to show that every T!
space Y can be embedded in F 7 for some m.
c) Let D denote the discrete space Ö0,1×. Use the theorem to show that every X" -space ]
satisfying dimÐ] Ñ œ ! (see Exercise E14) can be embedded in D7 for some m.
Parts b) and c) are due to Alexandroff.
Mrowka also proved that there is no T" -space \ such that every T" -space ] can be
embedded in X 7 for some m.
E19Þ Let \ œ Ö!ß "× and consider the product space \ 7 , where 7 is an infinite cardinal.
267
a) Show that \ 7 contains a discrete subspace of cardinality 7.
b) Show that AÐ\ 7 Ñ œ 7 Ðsee Example 4.6).
E20. A space \ is called totally disconnected every connected subset E satisfies lEl Ÿ "Þ Prove that
a totally disconnected compact Hausdorff space is homeomorphic to a closed subspace of Ö!ß "×7 for
some 7. (Hint: see Lemma V.5.6).
E21. Suppose \ is a countable space that does not have a countable neighborhood base at the point
+ − \Þ (For instance, let + œ Ð!ß !Ñ in the space P, Example III.9.8.)
i!
Let E4 œ ÖB − \ i! À B3 œ + for 3  4× and E œ ∞
4œ! E4 © \ . Prove that no point in the
(countable) space E has a countable neighborhood base. (Note: it is not necessary that \ be
countable. That condition simply forces E to be countable and makes the example more dramatic.)
268
5. The Quotient Topology
Suppose that for each α − E we have a map 1α À \α Ä ] , where \α is a topological space and ] is a
set. Certainly there is a topology for ] that will make all the 1α 's continuous: for example, the trivial
topology on ] . But what is the largest topology on ] that will do this? Let
g œ ÖS © ] À for all α − E, 1α" ÒSÓ is open in \α ×
It is easy to check that g is a topology on ] . For each α, by definition, each set 1α" ÒSÓ is open in \α
 so g makes all the 1α 's continuous. Moreover, if F © ] and F Â g , then for at least one α,
1α" ÒFÓ is not open in \α  so adding F to g would “destroy” the continuity of at least one 1α Þ
Therefore g is the largest possible topology on ] making all the 1α 's continuous.
Definition 5.1 Suppose 1α À \α Ä ] for each α − E. The strong topology g on ] generated by the
maps 1α is the largest topology on ] making all the maps 1α continuousß and S − g iff 1α" ÒSÓ is open
in \α for every α.
The strong topology generated by a collection of maps Ö1α À α − E× is “dual” to the weak topology in
the sense that it involves essentially the same notation but with “all the arrows pointing in the opposite
direction.” For example, the following theorem states that a map 0 out of a space with the strong
topology is continuous iff each map 0 ‰ 1α is continuous; but a map 0 into a space with the weak
topology generated by mappings 0α is continuous iff all the compositions 0α ‰ 0 are continuous (see
Theorem 2.6)
Theorem 5.2 Suppose ] has the strong topology generated by a collection of maps Ö1α À α − E×Þ If
^ is a topological space and 0 À ] Ä ^ , then 0 is continuous iff 0 ‰ 1α À \α Ä ^ is continuous for
each α − E.
1α
0
Proof For each α − E, we have \α Ä ] Ä ^ , and the 1α 's are continuous since ] has the strong
topology.
If 0 is continuous, so is each composition 0 ‰ 1α Þ
Suppose each 0 ‰ 1α is continuous and that Y is open in ^ . We want to show that 0 " ÒY Ó
is open in ] . But ] has the strong topology, so 0 " ÒY Ó is open in ] iff each 1α" Ò0 " ÒY ÓÓ
is open in \α Þ But 1α" Ò0 " ÒY ÓÓ œ Ð0 ‰ 1α Ñ" ÒY Ó which is open because 0 ‰ 1α is
continuous. ñ
We introduced the idea of the strong topology as a parallel to the definition of weak topology.
However, we are going to use the strong topology only in a special case: when there is only one map
1α œ 1 and 1 is onto.
Definition 5.3 Suppose Ð\ß g Ñ is a topological space and 1 À \ Ä ] is onto. The strong topology on
] generated by 1 is also called the quotient topology on ] . If ] has the quotient topology from 1, we
269
say that 1 À \ Ä ] is a quotient mapping and we say ] is a quotient of \ . We also say the ] is a
quotient space of \ and sometimes as ] œ \Î1.
From our earlier discussion we know that if \ is a topological space and 1 À \ Ä ] , then the quotient
topology on ] is g œ ÖY À 1" ÒY Ó is open in \×. Thus Y − g if and only if 1" ÒY Ó is open in \ :
notice in this description that
“only if”
“if”
guarantees that 1 is continuous and
guarantees that g is the largest topology on ] making 1 continuous.
Quotients of \ are used to create new spaces ] by “pasting together” (“identifying”) several points of
\ to become a single new point. Here are two intuitive examples:
i) Begin with \ œ Ò!ß "Ó and identify ! with " (that is, “paste” ! and " together to become a
single point). The result is a circle, W " Þ This identification is exactly what the map 1 À Ò!ß "Ó Ä W "
given by 1ÐBÑ œ Ðcos #1Bß sin #1BÑ accomplishes. It turns out (see below) that the usual topology on
W " is the same as quotient topology generated by the map 1. Therefore we can say that 1 is a quotient
map and that W " is a quotient of Ò!ß "Ó
ii) If we take the space \ œ W " and use a mapping 1 to “identify” the north and south poles
together, the result is a “figure-eight” space ] . The usual topology on ] (from ‘# ) turns out to be the
same as quotient topology generated by 1 (see below). Therefore we can say that 1 is a quotient
mapping and the the “figure-eight” is a quotient of W " .
Suppose we are given an onto map 1 À Ð\ß g Ñ Ä Ð] ß g w Ñ. How can we tell whether 1 is a quotient
map  that is, how can we tell whether g w is the quotient topology? By definition, we must check that
Y − g w iff 1" ÒY Ó − g Þ Sometimes it is fairly straightforward to do this. But the following theorem
can sometimes make things much easier.
Theorem 5.4 Suppose 1 À Ð\ß g Ñ Ä Ð] ß g w Ñ is continuous and onto. If 1 is open (or closed), then
g w is the quotient topology, so 1 is a quotient map. In particular, if \ is compact and ] is Hausdorff,
1 is a quotient mapping.
Note: Whether the map 1 À \ Ä ] is continuous depends, of course, on the topology g w , but if g w
makes 1 continuous, then so would any smaller topology on ] . The theorem tells us that if 1 is
both continuous and open (or closed), then g w is completely determined by 1: g w is the largest
topology making 1 continuous.
Proof Suppose Y © ] Þ W/ must show Y − g w iff 1" ÒY Ó − g Þ If Y − g w , then 1" ÒY Ó − g
because 1 is continuous. On the other hand, suppose 1" ÒY Ó − g . Since 1 is onto, 1Ò1" ÒY ÓÓ œ Y and
so, if 1 is open, Y œ 1Ò1" ÒY ÓÓ − g w Þ
For J © ] ß 1" Ò]  J Ó œ \  1" ÒJ ÓÞ It follows easily that J is closed in the quotient
space if and only if 1" ÒJ Ó is closed in \Þ With that observation, the proof that a continuous, closed,
onto map 1 is a quotient map is exactly parallel to the proof when 1 is open.
If \ is compact and ] is X# , then 1 must be closed, so 1 is a quotient mapping. ñ
Note: Theorem 5.4 implies that the map 1ÐBÑ œ Ðcos #1Bß sin #1BÑ from Ò!ß "Ó to W " is a quotient
map, but 1 is not open. The same formula 1 can be used to define a quotient map 1 À ‘ Ä W "
which is not closed (why?). Exercise E25 gives examples of quotient maps 1 that are neither
270
open nor closed.
Suppose µ is an equivalence relation on a space \ . For each B − \ , the equivalence class of B is
ÒBÓ œ ÖD − \ À D µ B×. The equivalence classes partition \ into a collection of nonempty pairwise
disjoint sets. (Conversely, it is easy to see that any partition of \ is the collection of equivalence
classes for some equivalence relation  namely, B µ D iff B and D are in the same set of the partition.)
The set of equivalence classes, ] œ ÖÒBÓ À B − \×, is sometimes denoted \Î µ Þ There is a natural
onto map 1 À \ Ä ] given by 1ÐBÑ œ ÒBÓ. We can think of the elements of ] as “new points” which
arise from “identifying together as one” all the members of each equivalence class in \ . If \ is a
topological space, we can give the set of equivalence classes ] the quotient topology. Conversely,
whenever 1 À \ Ä ] is any onto mapping, we can think of ] as the set of equivalence classes for
some equivalence relation on \  namely B µ C Í C − 1" ÐBÑ Í 1ÐCÑ œ 1ÐBÑ .
Example 5.5 For +, , − ™, define + µ , iff ,  + is even. There are two equivalence classes
Ò!Ó œ ÖÞÞÞß  %ß  #ß !ß #ß %ß ÞÞÞ× and Ò"Ó œ ÖÞÞÞ  $ß  "ß "ß $ß ÞÞÞ× so ™Î µ œ ÖÒ!Óß Ò"Ó×Þ
Define 1 À ™ Ä ™Î µ by 1Ð+Ñ œ Ò+Ó and give ™Î µ the quotient topology. A set Y is
open in ™Î µ iff 1" ÒY Ó is open in ™, and that is true for every Y © ™Î µ because ™ is discrete.
Therefore the quotient ™Î µ is a two point discrete space.
Example 5.6 Let Ð\ß .Ñ be a pseudometric space and define a relation µ in \ by B µ D iff
.ÐBß DÑ œ !. Let ] œ \Î µ , define 1 À \ Ä ] by 1ÐBÑ œ ÒBÓ. Give ] the quotient topologyÞ
Points at distance ! in \ have been “identified with each other” to become one point (the equivalence
class) in ] Þ
For ÒBÓß ÒDÓ − ] , define . w ÐÒBÓß ÒDÓÑ œ .ÐBß DÑ. In order to see that . w is well-defined,
we need to check that the definition is independent of the representatives chosen from the equivalence
classes:
If ÒB w Ó œ ÒBÓ and ÒD w Ó œ ÒDÓ, then .ÐBß B w Ñ œ ! and .ÐDß D w Ñ œ !Þ Therefore
.ÐBß DÑ Ÿ .ÐBß B w Ñ  .ÐB w ß D w Ñ  .ÐD w ß DÑ œ .ÐB w ß D w Ñ, and similarly
.ÐB w ß D w Ñ Ÿ .ÐBß DÑ. Thus .ÐB w ß D w Ñ œ .ÐBß DÑ so . w ÐÒB w Óß ÒD w ÓÑ œ . w ÐÒBÓß ÒDÓÑÞ
It is easy to check that . w is a pseudometric on ] . In fact, . w is a metric: if . w ÐÒBÓß ÒDÓÑ œ !, then
.ÐBß DÑ œ !, which means that B µ D and ÒBÓ œ ÒDÓÞ
We now have two definitions for topologies on ] : the quotient topology g and the metric topology
g. w . We claim that g œ g. w Þ To see this, first notice that
w
ÒCÓ − F%. ÐÒBÓÑ iff . w ÐÒCÓß ÒBÓÑ  % iff .ÐCß BÑ  % iff C − F%. ÐBÑ
w
w
Therefore 1" ÒF%. ÐÒBÓÑÓ œ F%. ÐBÑ and 1 ÒF%. ÐBÑÓ œ F%. ÐÒBÓÑ. But then we have
Y − g. w
iff
iff
iff
iff
Y is a union of . w -balls
1" ÒY Ó is a union of . -balls
1" ÒY Ó is open in \
Y −gÞ
271
The metric space Ð] ß . w Ñ is called the metric identification of the pseudometric space Ð\ß .Ñ. In effect,
we turn the pseudometric space into a metric space by agreeing that points in \ at distance ! are
“identified together” and treated as one point.
Note: In this particular example, it is easy to verify that the quotient mapping 1 À \ Ä ] is open,
so 1 would be a homeomorphism if only 1 were "  ". If the original pseudometric . is actually a
metric, then 1 is "  " and a homeomorphism: the metric identification of a metric space Ð\ß .Ñ is
itself.
Example 5.7 Exactly what does it mean if we say “identify the endpoints of Ò!ß "Ó and get a circle”?
Of course, one could simply take this to be the definition of a (topological) “circle.” Or, it could mean
that we already know what a circle is and are claiming that a certain quotient space is homeomorphic to
a circle. We take the latter point of view.
Let 1 À Ò!ß "Ó Ä W " be given by 1ÐBÑ œ Ðcos #1Bß sin #1B)Þ This map is onto and "  " except
that 1Ð!Ñ œ 1Ð"Ñ, so 1 corresponds to the equivalence relation on \ for which ! µ " (and there are no
other equivalences except that B µ B for every BÑÞ We can think of the equivalence classes as
corresponding in a natural way to the points of W " .
Here W " has its usual topology and 1 is continuous. However, since \ is compact and W " is Hausdorff,
Theorem 5.4 gives that the usual topology on W " is the quotient topology and 1 is a quotient map.
When it “seems apparent” that the result of making certain identifications produces some familiar
space ] , we need to check that the familiar topology on ] is actually the quotient topology. Example
5.7 is reassuring: if we believed, intuitively, that the result of identifying the endpoints of Ò!ß "Ó should
be W " but then found that the quotient topology on the set W " turned out to be different from the usual
topology, we would be inclined to think that we had made the “wrong” definition for a “quotient.”
Example 5.8 Suppose we take a square Ò!ß "Ó# and identify points on top and bottom edges using the
equivalence relation ÐBß !Ñ µ ÐBß "ÑÞ We can schematically picture this identification as
272
The arrows indicate that the edges are to be identified as we move along the top and bottom edges in
the same direction. We have an obvious map 1 from Ò!ß "Ó# to a cylinder in ‘$ which identifies points
in just this way, and we can think of the equivalence classes as corresponding in a natural way to the
points of the cylinder.
The cylinder has its usual topology from ‘$ and the map 1 is (clearly) continuous and onto. Again,
Theorem 5.4 gives that the usual topology on the cylinder is, in fact, the quotient topology.
273
Example 5.9 Similarly, we can show that a torus (the “surface of a doughnut”) is the result of the
following identifications in Ò!ß "Ó# : ÐBß !Ñ µ ÐBß "Ñ and Ð!ß CÑ µ Ð"ß CÑ
Thinking in two steps, we see that the identification of the two vertical edges produces a cylinder; the
circular ends of the cylinder are then identified (in the same direction) to produce the torus.
The two circles darkly shaded on the surface represent the identified edges.
We can identify the equivalence classes naturally with the points of this torus in ‘$ and just as before
we see that the usual topology on the torus is in fact the quotient topology.
274
Example 5.10 Define an equivalence relation µ in Ò!ß "Ó# by setting ÐBß !Ñ µ Ð"  Bß "ÑÞ Intuitively,
the idea is to identify the points on the top and bottom edges with each other as we move along the
edges in opposite directions. We can picture this schematically as
Physically, we can think of a strip of paper and glue the top and bottom edges together after making a
“half-twist.” The quotient space \Î µ is called a Mobius
strip.
¨
We can take the quotient \Î µ as the definition of a Mobius
strip, or we can consider a “real” Mobius
¨
¨
strip Q in ‘$ and define a map 1 À Ò!ß "Ó# Ä Q that accomplishes the identification we have in mind.
In that case there is a natural way to associate the equivalence classes to the points of the torus in ‘$
and again Theorem 5.4 guarantees that the usual topology on the Mobius
strip is the quotient topology.
¨
275
Example 5.11 If we identify the vertical edges of Ò!ß "Ó# (to get a cylinder) and then identify its
circular ends with a half-twist (reversing orientation): Ð!ß CÑ µ Ð"ß CÑ and ÐBß !Ñ µ Ð"  Bß "Ñ. We get
a quotient space which is called a Klein bottle.
It turns out that a Klein bottle cannot be embedded in ‘$  the physical construction would require a
“self-intersection” (additional points identified) which is not allowed. A pseudo-picture looks like
In these pictures, the thin “neck” of the bottle actually intersects the main body in order to re-emerge
“from the inside”  in a “real” Klein bottle (in ‘% , say), the slef-intersection would not happen.
In fact, you can imagine the Klein bottle as a subset of ‘% using color as a 4>2 dimension. To each
point ÐBß Cß DÑ on the “bottle” pictured above, add a 4th coordinate to get ÐBß Cß Dß <ÑÞ Now color the
points on the bottle in varying shades of red and let < be a number measuring the “intensity of red
coloration at a point.” Do the coloring in such a way that the surface “blushes” as it intersects itself 
so that the points of “intersection” seen above in ‘$ will be different (in their 4th coordinates).
Alternately, you can think of the Klein bottle as a parametrized surface traced out by a moving point
T œ ÐBß Cß Dß >Ñ where Bß Cß D depend on time > and > is recorded as a 4th -coordinate. A point on the
surface then has coordinates of form ÐBß Cß Dß >Ñ. At “a point” where we see a self-intersection in ‘$ ß
there are really two different points (with different time coordinates >).
276
Example 5.12
1) In W " , identify antipodal points  that is, in vector notation, T µ  T for each T − W " .
Convince yourself that the quotient W " Î µ is W " .
2) Let H# be the unit disk ÖT − ‘# À lT l Ÿ "×. Identify antipodal points on the boundary of
H : that is, T µ  T if T − W " © H# Þ The quotient HÎ µ is called the projective plane, a space
which, like the Klein bottle, cannot be embedded in ‘$ .
#
3) For any space \ , we can form the product \ ‚ Ò!ß "Ó and let ÐBß "Ñ µ ÐCß "Ñ for all
Bß C − \ . The quotient Ð\ ‚ Ò!ß "ÓÑÎ µ is called the cone over \ . (Why?)
4) For any space \ , we can form the product \ ‚ Ò  "ß "Ó and define ÐBß "Ñ µ ÐCß "Ñ and
ÐBß  "Ñ µ ÐCß  "Ñ for all Bß C − \ . The quotient Ð\ ‚ Ò  "ß "ÓÑÎ µ is called the suspension of \ .
(Why?)
There is one other very simple construction for combining topological spaces. It is often used in
conjunction with quotients.
Definition 5.13 For each α − E, let Ð\α ß gα Ñ be a topological space, and assume that the sets \α are
pairwise disjoint. The topological sum (or “free sum”) of the \α 's is the space Ðα−E \α ß g Ñ where
g œ ÖS © α−E \α À S ∩ \α is open in \α for every α − E}. We denote the topological sum by
 \α . In the case lEl œ #, we use the simpler notation \"  \# Þ
α−E
In  \α , each \α is a clopen subspace. Any set open (or closed) in \α is open (or closed) in the sum.
The topological sum  \α can be pictured as a union of the disjoint pieces \α , all “far apart” from
α−E
α−E
each other  so that there is no topological “interaction” between the pieces.
Example 5.14 In ‘# , let E" and E# be open disks with radius " and centers at Ð!ß !Ñ and Ð$ß !ÑÞ
Then toppological sum E"  E# is the same as E" ∪ E# with subspace topology
Let F" be an open disk with radius " centered at Ð!ß !Ñ and let F# be a closed disk
with radius " centered at Ð#ß !Ñ. Then F"  F# is not the same as F" ∪ F# with the subspace topology
(why? ÑÞ
Exercise 5.15 Usually it is very easy to see whether properties of the \α 's do or do not carry over to
 \α . For example, you should convince yourself that:
α−E
1) If the \α 's are nonempty and separable, then  \α is separable iff lEl Ÿ i! .
2) If the \α 's are nonempty and second countable, then  \α is second countable iff lEl Ÿ i! .
α−E
3) A function 0 À  \α Ä ^ is continuous iff each 0 l\α is continuous.
α−E
α−E
277
4) If .α Ÿ 1 is a metric for \α , then the topology on  \α is the same as g. where
α−E
. ÐBß CÑ if Bß C − \α
.ÐBß CÑ œ  α
"
otherwise
so  \α is metrizable if all the \α 's are metrizable. You should be able to find similar statements for
α−E
other topological properties such as first countabe, second countable, Lindelof,
¨ compact, connected,
path connected, completely metrizable, ... .
Definition 5.16 Suppose \ and ] are disjoint topological spaces and 0 À E Ä ] , where E © \ . In
the sum \  ] , define B µ C iff C œ 0 ÐBÑÞ If we form Ð\  ] ÑÎ µ , we say that we have attached
\ to ] with 0 and write this space as \  0 ] .
For each : − 0 ÒEÓß its equivalence class under µ is Ö:× ∪ 0 " ÒÖ:×ÓÞ You may think of the function
“attaching” the two spaces by repeatedly selecting a group of points in \ , identifying them together,
and “sewing” them all onto a single point in ]  just as you might run a needle and thread through
several points in the fabric \ and then through a point in ] and pull everything tight.
Example 5.17
1) Consider disjoint cylinders \ and ] . Let E be the circle forming one end of \ and F the
circle forming one end of ] . Let 0 À E Ä 0 ÒEÓ œ F © ] be a homeomorphism. Then 0 “sews
together” \ and ] by identifying these two circles. The result is a new cylinder.
2) Consider a sphere W # © ‘$ . Excise from the surface W # two disjoint open disks H" and H#
and let E" ∪ E# be union of the two circles that bounded those disks. Let ] be a cylinder whose
ends are bounded by the union of two circles, F" ∪ F# . Let 0 be a homeomorphism carrying the
points of E" and E" clockwise onto the points of F" and F# respectively..
Then W #  0 ] is a “sphere with a handle.”
3) Consider a sphere W # © ‘$ . Excise from the surface W # an open disk and let E be the
circular boundary of the hole in the surface W # . Let Q be a Mobius
strip and let F be the curve that
¨
"
bounds it. Of course, F ¶ W . Let 0 À E Ä F be a homeomorphism. The we can use 0 to join the
spaces by “sewing” the edge of the Mobius
strip to the edge of the hole in W # . The result is a “sphere
¨
with a crosscap.”
There is a very nice theorem, which we will not prove here, which uses all these ideas. It is a
“classification” theorem for certain surfaces.
Definition 5.18 A Hausdorff space \ is a 2-manifold if each B − \ has an open neighborhood Y that
is homeomorphic to ‘# . Thus, a 2-manifold looks “locally” just like the Euclidean plane. A surface is
a Hausdorff 2-manifold.
Theorem 5.19 Let \ be a compact, connected surface. The \ is homeomorphic to a sphere W # or to
W # with a finite number of handles and crosscaps attached.
You can read more about this theorem and its proof in Algebraic Topology: An Introduction (William
Massey).
278
Exercises
E22. a) Let µ be the equivalence relation on ‘# given by ÐB" ß C" Ñ µ ÐB# ß C# Ñ iff C" œ C# . Prove that
‘# Î µ is homeomorphic to ‘.
b) Find a counterexample to the following assertion: if µ is an equivalence relation on a
space \ and each equivalence class is homeomorphic to the same space ] , then Ð\Î µ Ñ ‚ ] is
homeomorphic to \ .
Why might someone ever wonder whether this assertion might be true? In part a), we have
\ œ ‘ ‚ ‘, each equivalence class is homeomorphic to ‘ and (\Î µ Ñ ‚ ‘ ¶ ‘ ‚ ‘ ¶ \ . In this
example, you “divide out” equivalence classes that all look like ‘, then “multiply” by ‘, and you're
back where you started.
c) Let 1 À ‘# Ä ‘ be given by 1ÐBß CÑ œ B#  C# .
homeomorphic to what familiar space?
Then the quotient space ‘# Î1 is
d) On ‘# , define an equivalence relation (B" ß C" Ñ µ ÐB# ß C# Ñ iff B"  C"# œ B#  C## . Prove
that ‘ / µ is homeomorphic to some familiar space.
#
E23. For Bß C − Ò!ß "Ó, define B µ C iff B  C is rational. Prove that the corresponding quotient
space Ò!ß "ÓÎ µ is trivial.
E24. Prove that a 1-1 quotient map is a homeomorphism.
E25. a) Let ]" œ Ò!ß "Ó with its usual topology and ]# œ Ò!ß "Ó with the discrete topology. Define
1 À ]"  ]# Ä ] by letting 1 be “the identity map” on both ]" and ]# Þ Prove that 1 is a quotient map
that is neither open nor closed.
b) Let ‘# have the topology g for which a subbase consists of all the usual open sets together
with the singleton set ÖÐ!ß !Ñ×. Let ‘ have the usual topology and let 0 À ‘# Ä ‘ be the projection
0 ÐBß CÑ œ B. Prove that 0 is a quotient map which is neither open nor closed.
E26. State and prove a theorem of the form:
“for two disjoint subsets E and F of ‘# , E  F ¶ E ∪ F iff ... ”
E27. Let ‘# have the topology g for which a subbasis consists of all the usual open sets together with
the singleton set ÖÐ!ß !Ñ×. Let ‘ have the usual topology and define 0 À ‘# Ä ‘ by 0 ÐBß CÑ œ B.
Prove that 0 is a quotient map which is neither open nor closed.
E28. Let ]" œ   Ð ‚ Ð!ß "ÑÑ and let ]# œ ]"  Ò!ß "Ñ. Prove ]" is not homeomorphic to ]# but
279
that each is a continuous one-to-one image of the other
E29. Show that no continuous image of ‘ can be represented as a topological sum \  ] , where
\ß ] Á g. How can this be result be strengthened?
E30. Suppose \= (s − W ) and ]> (> − X ) are pairwise disjoint spaces. Prove that  \= ‚  ]> is
homeomorphic to
 Ð\= ‚ ]> ÑÞ
>−X
=−W
=−Wß>−X
E31. This problem outlines a proof (due to Ira Rosenholtz) that every nonempty compact metric space
\ is a continuous image of the Cantor set G . From Example 4.5, we know that \ is homeomorphic to
a subspace of Ò!ß "Ói! Þ
a) Prove that the Cantor set G © ‘ consists of all reals of the form 
∞
4œ!
+4
$4
where
each +4 œ ! or 2.
b) Prove that Ò!ß "Ó is a continuous image of G . Hint: Define 1Ð
∞
4œ!
+4
$4
Ñœ
∞
4œ!
+4
#4
Þ
c) Prove that the cube Ò!ß "Ói! is a continuous image of G .
Hint: By Corollary 2.21, G ¶ Ö!ß #×i! ¶ G i!. Use 1 from part b) to define 0 À G i! Ä Ò!ß "Ói! by
0 ÐB" ß B# ß ÞÞÞß ÞÞÞÑ œ Ð1ÐB" Ñß 1ÐB# Ñß ÞÞÞß ÞÞÞÑ
d) Prove that a closed set O © G is a continuous image of G .
(Hint: G is homeomorphic to the “middle two-thirds" set G w consisting of all reals of the form
∞
 ,44 . G w has the property that if Bß C − G w , then B  C Â G w Þ If O w is closed in G w , we can map
4œ!
w
'
#
G Ä O by sending each point B to the point in O w nearest to B.)
e) Prove that every nonempty compact metric space \ is a continuous image of G .
.
280
Chapter VI Review
Explain why each statement is true, or provide a counterexample.
1. Ð!ß "Ñi! is open in Ò!ß "Ói! Þ
2. Suppose J is a closed set in Ò!ß "Ó ‚ ‘. Then 1# ÒJ Ó is a closed set in ‘Þ
3. i! is discrete
4. If G is the Cantor set, then there is a complete metric on G i! which produces the product topology.
5. Let ‘‘ have the box topology. A sequence Ð08 Ñ Ä 0 − ‘‘ iff Ð08 Ñ Ä 0 uniformly.
6. Let 08 À Ò!ß "Ó Ä Ò!ß "Ó be given by 08 ÐBÑ œ B8 . The sequence Ð08 Ñ has a limit in Ò!ß "ÓÒ!ß"Ó .
7. Let 1 − ‘‘ be defined by 1ÐBÑ œ B# for all B − ‘. Give an example of a sequence Ð08 Ñ of distinct
functions in ‘‘ that converges to 1.
8. Let G be the Cantor set. Then G is homeomorphic to the topological sum G  G .
9. The projection maps 1B and 1C from ‘# Ä ‘ separate points from closed sets.
10. The letter N is a quotient of the letter M .
11. Suppose µ is an equivalence relation on \ and that for B − \ß Ò B Ó represents its equivalence
class. If B is a cut point of \ , then Ò B Ó is a cut point of the quotient space \Î µ .
12. If 1À \ Ä ] is a quotient map and ] is compact T# , then ] is compact T# .
13. Suppose E is infinite and, that in each space \α (α − E) there is a nonempty proper open subset
Sα . Then Sα is not a basic open set of the product \ œ \α , and, in fact, Sα cannot even be
open in the product.
∞
14. If \ œ ∞
8œ" E8 , where the E8 's are disjoint clopen sets in \ , then \ z
8œ" E8
topological sum of the E8 's).
( œ the
15. Let \8 œ Ö!ß "× and ]8 œ  with their usual topologies. Then  \8 is homeomorphic to  ]8 Þ
∞
∞
8œ"
8œ"
16. Let \8 œ Ö!ß "× and ]8 œ  with their usual topologies. Then ∞
8œ" \8 is homeomorphic to
∞
8œ" ]8 Þ
17. Let E œ ÖÐBß Cß DÑ − ‘$ À B  C#  #CD  D #  lsinÐBCDÑl ×, and let 0 À ‘$ Ä ‘ be given by
0 ÐBß Cß DÑ œ B  #. Then 0 ÒEÓ is open but not closed in ‘.
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∞
18. Suppose E8 is a connected subset of \8 Ð Á gÑ and that ∞
8œ" E8 is dense in
8œ" \8 Þ Then each
\8 is connected.
19. In ‘‘ , every neighborhood of the function sin contains a step function (that is, a function with
finite range).
20. Let X be an uncountable set with the cocountable topology. Then {ÐBß BÑ À B − \× is a closed
subset of the product \ ‚ \Þ.
21. Let E œ Ö 8" À 8 − × © ‘, and let 7 be any cardinal number. E5 is a closed set in ‘5 Þ
22. If G is the Cantor set, then G ‚ G ‚ G ‚ G is homeomorphic to G ‚ G .
33. W " ‚ W " is homeomorphic to the “infinity symbol”: ∞
34. 31. Let T be the set of all real polynomials in one variable, with domain restricted to Ò!ß "Ó, for
which ranÐT Ñ © Ò!ß "ÓÞ Then T is dense in Ò!ß "ÓÒ!ß"Ó Þ
35. Every metric space is a quotient of a pseudometric space.
37. A separable metric space with a basis of clopen sets is homeomorphic to a subspace of the Cantor
set.
38. Let {\α À α − E} be a nonempty product space. Each \α is a quotient of the product space
Ö\α À α − E×
-
39. Suppose \ does not have the trivial topology. Then \ # cannot be separable.
40. Every countable space \ is a quotient of 
41.  ‚  is homeomorphic to the sum of i! disjoint copies of .
42. Suppose B œ ÐBα Ñ − int E, where E © \α . Then for every α, Bα − int 1α ÒEÓ.
43. The unit circle, W " , is homeomorphic to a product α−E \α , where each \α © Ò!ß "Ó
(i.e., W " can be “factored” into a product of subspaces of Ò!ß "Ó ).
44. i! is homeomorphic to ‘.
282