Lecture Notes on Set Theory and Point-Set Topology
Jack F. McMillan1
1
Hawaii Pacific University, College of Natural and Computational Sciences
These notes constitute a foundation for a possible course on set theory and point-set topology
with an eye toward differential geometry and its applications in the physical sciences. This paper is
pedagogical in nature and does not represent new research. These notes will be updated and thus
should be considered incomplete. Hopefully students will find this material useful but as always
readers should consult a number of resources.
There are more things in heaven and earth,
Horatio, Than are dreamt of in your philosophy.
William Shakespeare, Hamlet
I.
INTRODUCTION.
The history of mathematics and physics have long been intertwined. Physicists have provided contributions to
mathematics directly (Newton and Bessel to name but a few). Mathematicians have provided a language for physics
to be expressed. The role of mathematics in the development of physics cannot be overemphasized. Whilst the goals
of the two fields differ, modern physics as we know it could not exist without the powerful tools mathematics provides.
II.
SET THEORY.
It is common practice in mathematics today to categorize the various entities we encounter such as numbers,
functions, matrices, vectors etc into groupings known as sets. So what exactly is a set? How do we determine set
membership? Can we form new sets from old ones? Answers to these and other pressing issues are the primary topics
of this chapter. It should be noted that we will be using Zermelo-Fraenkel (ZF) set theory to avoid some logical
pitfalls such as self-referencing sets which lead to paradoxes (for example the famous one devised by Bertrand Russell
in 1902 - ”the set of all sets” paradox which led to the idea of classes as distinct objects from sets). Furthermore, if I
refer to a universal set, it is to be understood in the context of simply being some sufficiently large set that includes
as subsets any sets I may be currently discussing. Whence I will not require Zermelo’s Axiom of Comprehension and
shall at all times assume restricted comprehension.
Definition of a set.
A set is quite simply a collection of objects. The number of objects contained within a set may be finite, countably
infinite ( e.g., the set of natural numbers 1, 2, 3, ... ) or uncountably infinite (e.g., the set of real numbers). We
normally use curly brackets { and } to indicate a set. Some special sets have their own symbols. Let’s look at some
examples
{1,2,3} - the set containing the numbers 1, 2, and 3.
N1 = {1,2,3,...} - the set of all natural numbers.
N0 = {0,1,2,3,...} - the set of whole numbers.
Z = { ... -3, -2, -1, 0, 1, 2, 3, ... } - the set of integers.
Q - the set of rational numbers.
P - the set of irrational numbers.
2
R - the set of real numbers.
C - the set of complex numbers.
H - the set of quaternions.
O - the set of octonions.
∅ = {} - the null or empty set, a set with no elements.
Sometimes we define a set by stating the properties required for set membership. This format is called set-builder
notation and has the general form:
{ x | condition A(x) is true } - the set of all x-values which satisfies A.
For example, { x ∈ R | x2 = 1 } = {−1, 1}.
As another example, consider the set of complex numbers mentioned earlier. In set-builder notation we could write
C = { x + i y | x , y ∈ R }.
Now that you have an understanding of sets, let’s look at ways to create new sets from old. First we define a subset.
A subset X of a set Y is one in which every element of X is also in Y. For example let Y = {1,2,3}. Possible subsets
include {1} and {2,3}. In fact all subsets of Y can be easily delineated:
∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, and {1,2,3} = Y.
Note both the null set and the entire set Y itself are considered subsets. However a proper subset is one that is not
equal to the entire set. So Y is not a proper subset. On the other hand, the null set is considered proper. We identify
subsets using the notation X ⊂ Y, meaning X is a subset of Y. If X is permitted to include Y, we write X ⊆ Y. One
also occasionally sees Y ⊃ X or Y ⊇ X, meaning Y is a superset of X, implying Y contains everything in X and
possibly more. There is a new set we can now form - the power set. The power set of a set Y contains all its subsets.
We denote the power set by P. In the above example, P(Y ) = { ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }.
Notice the power set has 8 = 23 elements in this case, each of which comprises a set in its own right. This will always
be the case - a set of N elements forms a power set containing 2N elements. (Do you see why? Hint: to form a subset,
we need to ask if an element x is in that subset. The answer is binary, yes or no, whence 2N ). Some mathematicians
even denote the power set of Y by 2Y ; this author however prefers the ’P’ notation which traces its lineage to the
writings of Weierstrauß.
III.
BINARY OPERATIONS ON SETS. UNION AND INTERSECTION.
The union of sets. Given sets A and B, we define the union of A and B A ∪ B as the set containing all elements
found in both A and B. For example, let A = {1, 2, 3, 4} and B = {1, 2, 10, 13, 27}. Then A ∪ B = {1, 2, 3, 4, 10, 13, 27}.
Note that 1 and 2 are common to both sets A and B but appear only once the union - we do not double-count set
elements. So if we have a set, say C = {1, 2, 3, 3} for instance, this would just be written as C = {1, 2, 3}. The extra
3 is not included.
The intersection of sets. The intersection of two sets is the set containing only their mutual elements. We denote
intersection by ∩, the intersection of sets A and B is then A ∩ B. For example, let A = {1, 2, 4, 8, 16} and B =
{4, 16, 25}. The A ∩ B = {4, 16}. If two sets have no common elements, their intersection is the null set. For example,
if X = {5, 25, 125} and Y = {3, 9, 27} then X ∩ Y = ∅.
3
Now that union and intersection have been defined, we should note the operations are not limited to merely
Sn two
sets. Say I have n number of sets, A1 , A2 , A3 , . . . An . The set consisting of all elements
can
be
written
as
i=1 Ai .
Tn
Likewise the set containing only those elements common to all sets Ai is denoted by i=1 Ai . In some situations we
might desire only particular sets.
might consider the dummy index to be taken from a so-called ’index set’.
S Then we T
In that event we would write j∈I Aj or j∈I Aj for union and intersection, respectively. Indeed, the index set I
S
T
might be continuous rather than discrete, in which case we would write j⊆I Aj and j⊆I Aj .
After that brief digression, let’s return to the case of two sets. In set-builder notation, A ∪ B = { x | x ∈
A or x ∈ B }. Of course x can be in both A and B, so we’re not using the ‘exclusive or’ here but rather the ‘inclusive
or’, that is to say and/or. That is a fine point some texts fail to mention. In everyday life one typically employs the
‘exclusive or’ rather than the inclusive version found in set theory. For instance a pollster might ask whether you
support candidate A or candidate B. It is tacitly assumed the response won’t be “I support both”. Likewise, A ∩ B
= { x | x ∈ A and x ∈ B}. If the intersection of A and B is the null set A ∩ B = ∅, that is, they possess no common
elements we then say A and B are disjoint sets. If it is the case A ⊆ B and B ⊆ A, then A = B. This identity
relation is frequently used in solving proofs, as we shall soon see.
Venn Diagrams.
Sets may be visualized using Venn diagrams. They provide the student a useful tool, however they shouldn’t be
employed as a substitute for a formal proof. That said, let’s draw a simple Venn diagram:
A
B
Figure 1: The intersection A ∩ B.
We have two sets A and B and the shaded region represents A ∩ B. What if we wanted A ∪ B instead? Simple, we
merely shade both sets.
A
B
Figure 2: The union A ∪ B.
Notice the two sets A and B are enclosed within a box. This is not done merely for aesthetics, but rather to
emphasize the fact we shall always assume any sets under consideration are contained within some larger ’universal’
set.
Complements of Sets.
Given a set A, we define its complement Ac as the set containing only elements not found in A. Thus we write
A := {x ∈ X | x ∈
/ A}, where X denotes the universal set.
c
4
A
B
Figure 3: The set Ac is represented in the shaded region. B contains elements in both A and Ac .
DeMorgan’s Laws.
We are now in a position to state two important theorems developed by Augustus De Morgan (1806 -1871), a
British mathematician and logician. De Morgan’s laws can in fact be summarized succinctly in the statement ”union
and intersection interchange under complementation”. That is perhaps too terse, so we state them as follows:
• De Morgan’s First Law - the complement of the union equals the intersection of the complements. That is,
(A ∪ B)c = Ac ∩ B c .
• De Morgan’s Second Law - the complement of the intersection equals the union of the complements. That is,
(A ∩ B)c = Ac ∪ B c .
Proof of the First De Morgan Law.
Let x ∈ (A ∪ B)c . Then x ∈
/ A ∪ B. Thus (x ∈
/ A) ∧ (x ∈
/ B). This implies x ∈ Ac and x ∈ B c . Therefore
x ∈ Ac ∩ B c . We conclude (A ∪ B)c ⊆ Ac ∩ B c . Now let x ∈ Ac ∩ B c . Thus x ∈
/ A and x ∈
/ B. This implies x ∈
/ A∪B
thus x ∈ (A ∪ B)c . So Ac ∩ B c ⊆ (A ∪ B)c . Therefore Ac ∩ B c = (A ∪ B)c . Quod erat demonstrandum.
Proof of the Second De Morgan Law.
Let x ∈ (A ∩ B)c . Then x ∈
/ A ∩ B. Thus (x ∈
/ A) ∨ (x ∈
/ B). This implies x ∈ Ac or x ∈ B c . Therefore x ∈ Ac ∪ B c .
c
c
c
c
We conclude (A ∩ B) ⊆ A ∪ B . Now let x ∈ A ∪ B c . Thus x ∈
/ A or x ∈
/ B. This implies x ∈
/ (A ∩ B) thus
x ∈ (A ∩ B)c . So Ac ∪ B c ⊆ (A ∩ B)c . Therefore Ac ∪ B c = (A ∩ B)c . Quod erat demonstrandum.
Naturally we can generalize the laws to include arbitrary numbers of sets. Let I represent some indexing set. Then
the laws take the form:
S
T
• ( j∈I Aj )c = j∈I Acj
T
S
• ( j∈I Aj )c = j∈I Acj
In addition to De Morgan’s Laws there are some additional rules governing set algebra we should mention. Some
are trivial yet necessary for a self-consistent set theory and proofs will not be supplied. Others such as the distributive
laws however merit proof. We begin by noting the binary operations of union ∪ and intersection ∩ are commutative
and associative. In mathematical notation these rules are given as:
• Commutative Law - A ∪ B = B ∪ A and A ∩ B = B ∩ A.
• Associative Law - (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C).
5
Identity Laws
Let X be the universal set and let A ⊂ X be an arbitrary subset of X. The following identity rules hold:
• A ∪ ∅ = A.
• A ∩ X = A.
• A ∪ A = A.
• A ∩ A = A.
Complement Laws.
• X c = ∅.
• ∅c = X.
• A ∪ Ac = X.
• A ∩ Ac = ∅.
The Distributive Laws
We now consider the two distributive laws known as ’union over intersection’ and ’intersection over union’. I will
provide proofs as these laws are very important.
• Union Over Intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
• Intersection Over Union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Proof of Union Over Intersection.
Let x ∈ A ∪ (B ∩ C). Then (x ∈ A) ∨ (x ∈ B ∩ C). Let x ∈ A. Then x ∈ A ∪ B and x ∈ A ∪ C. Whence
x ∈ (A ∪ B) ∩ (A ∪ C). Now let x ∈ B ∩ C. So (x ∈ B) ∧ (x ∈ C). This implies (x ∈ B ∪ A) ∧ (x ∈ C ∪ A). We
conclude A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C). We now need to show A ∪ (B ∩ C) ⊇ (A ∪ B) ∩ (A ∪ C). Let
x ∈ (A ∪ B) ∩ (A ∪ C). Thus (x ∈ A ∪ B) ∧ (x ∈ A ∪ C). If x ∈ A, then x ∈ A ∪ (B ∩ C. If x ∈ B, then x ∈ C ⇒
x ∈ B ∩ C. This then implies x ∈ (B ∩ C) ∪ A. Therefore ∩ (A ∪ C) ⊆ A ∪ (B ∩ C). This completes the proof.
Quod erat demonstrandum.
Proof of Intersection Over Union.
Let x ∈ A ∩ (B ∪ C). Then (x ∈ A) ∧ (x ∈ B ∪ C). If x ∈ B, then x ∈ A ∩ B. Now let x ∈ C,
then x ∈ A ∩ C. Whence x ∈ (A ∩ B) or x ∈ A ∩ C). Thus x ∈ (A ∩ B) ∪ (A ∩ C). We conclude
A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C). Now let x ∈ (A ∩ B) ∪ (A ∩ C). Thus (x ∈ (A ∩ B)) ∨ (x ∈ (A ∩ C)).
Let x ∈ A ∩ B. Then (x ∈ A) ∧ (x ∈ B). This implies (x ∈ A) ∧ (x ∈ B ∪ C). Therefore x ∈ A ∩ (B ∪ C). If
x ∈ (A ∩ C), then we also obtain x ∈ A ∩ (B ∪ C). We surmise (A ∩ B) ∪ (A ∩ C) ⊆ A ∩ (B ∪ C). This
finishes the proof. Q.E.D.
The distributive laws may also be formulated to include families of sets. Denoting an indexing set by I, the laws
are then given by:
•B ∪
T
j∈I
Aj
=
S
j∈I
(B ∩ Aj ).
6
• B ∩
S
j∈I
Aj
=
T
j∈I
(B ∪ Aj ).
Difference of Sets.
We define the difference of two sets A and B as the set of elements contained within one set but not the other.
Specifically, A - B := { x ∈ A | (x ∈ A) ∧ (x ∈
/ B) }. We can express the difference then as A - B = A ∩ B c . One
may visualize the difference using a Venn diagram as seen below.
A
B
Figure 4: The set A - B = A ∩ B c .
Looking at the above diagram, it is apparent A - B must also equate to A ∩ (A ∩ B)c . This is easily verified using
the distributive laws:
A ∩ (A ∩ B)c = A ∩ (Ac ∪ B c ) = (A ∩ Ac ) ∪ (A ∩ B c ) = ∅ ∪ (A ∩ B c ) = A ∩ B c = A - B.
The Symmetric Difference.
We may conceptually extend the difference to form another set, the symmetric difference of sets. Denoted by A4B,
the symmetric difference (also known as the disjunctive union) is simply the union of A - B and B - A viz, A 4 B :=
(A - B) ∪ (B - A) = (A ∩ B c ) ∪ (B ∩ Ac ). We illustrate the symmetric difference in the Venn diagram below.
A
B
Figure 5: The shaded region illustrates the symmetric difference A 4 B.
We note the symmetric difference may also be expressed as (A ∩ (A ∩ B)c ) ∪ (B ∩ (A ∩ B)c ). Also note here we
may use the ‘exclusive or ’ in that x ∈ A 4 B implies x is either in A but not in B or x is in B but not in A. Thus
A 4 B can be considered the union of sets under the ‘exclusive or’.
Some Rules Regarding the Null Set.
As mentioned earlier, the null or empty set is that set containing no elements. It is denoted by ∅ which actually is
a Danish letter and not the Greek letter φ. The null set is a subset of every set and thus is a subset of itself: ∅ ⊆ ∅.
Now let us consider the following questions.
• Is ∅ ∈ ∅? No, the null set isn’t a member of itself. It has no elements.
7
• Is ∅ ⊆ ∅? Yes. It’s a subset of every set, including itself.
• Is ∅ = P(∅)? No. The power set of ∅ = { ∅ } =
6 ∅.
• Is ∅ ∈ { ∅ }? Yes, it is a member of this set.
• Is ∅ ⊆ { ∅ }? Yes, it’s a subset of every set.
• ∅ ∈ { ∅ } ∈ { {∅} }. Is ∅ ∈ { {∅} }? No, ∅ is not a member of { {∅} }. Therefore A ∈ B ∈ C does not imply
A ∈ C in general. Is ∅ ⊆ { {∅} }? Yes, the null set is a subset of every set.
Some Observations On Power Sets.
Given a set S, recall we defined the power set P(S) to be that set composed solely of subsets of S, i.e., P(S) =
{A | A ⊆ S }. We note power sets obey the following properties:
1. If A ⊆ B, then P(A) ⊆ P(B),
2. Given sets A and B, P(A) ∪ P(B) ⊆ P(A ∪ B),
3. Given sets A and B, P(A) ∩ P(B) = P(A ∩ B).
Proof of (1): This is easily demonstrated. Since A is a subset of B, every subset of A must also be a subset of B.
Whence the power set P(B) must contain all subsets of A. This implies P(B) ⊇ P(A).
Proof of (2): Let x ∈ P(A) ∪ P(B). Then x ⊆ A or x ⊆ B. If x ⊆ A, then x ⊆ ( A ∪ B ) ⇒ x ∈ P(A ∪ B).
Likewise if x ⊆ B then x must also satisfy x ⊆ ( A ∪ B ) ⇒ x ∈ P(A ∪ B). However it is not generally the case
P(A) ∪ P(B) = P(A ∪ B). To see why, consider two elements a ∈ A ∈
/ B and b ∈ B ∈
/ A. The set {a, b } ∈
P(A ∪ B), but is neither a subset of A nor of B and thus cannot be in P(A) ∪ P(B). In fact, equality only holds
when either A = B or one of the sets is ∅.
Proof of (3): Let x ∈ P(A) ∩ P(B). Then x ∈ P(A) and x ∈ P(B). Thus x ⊆ A and x ⊆ B. This
implies x ⊆ A ∩ B ⇒ x ∈ P(A ∩ B). Thus P(A) ∩ P(B) ⊆ P(A ∩ B). Now let x ∈ P(A ∩ B). Thus
x ⊆ A ∩ B ⇒ (x ⊆ A) ∧ (x ⊆ B) ⇒ (x ∈ P(A)) ∧ (x ∈ P(B)) ⇒ x ∈ P(A) ∩ P(B). Thus P(A ∩ B) ⊆
P(A) ∩ P(B). This completes the proof. Q.E.D.
Cardinality.
Given a set S, the cardinality of the set is simply the number of elements within the set. So, for instance, The set
A = { 10, 20, 30 } has a cardinality of 3 and we would write as Card(A) = 3. We can think of cardinality as a map
from a set to the natural numbers: Card: S → N1 . We can establish a rule governing the cardinalities of finite sets.
Theorem: Given sets A and B, Card(A ∪ B) = Card(A) + Card(B) - Card(A ∩ B). Equivalently, Card(A ∪ B)
= Card(A 4 B) + Card(A ∩ B).
Proof. Let Card(A) = m and Let Card(B) = n. If A and B are disjoint then Card(A ∪ B) is clearly just m +
n. However if A ∩ B 6= ∅, then adding cardinalities counts the intersection twice, so we must subtract Card(A ∩ B)
from m + n to account for the double-counting of the intersection. We can also express A ∪ B as the disjoint union
( A 4 B ) ∪ ( A ∩ B ). Since these sets are disjoint, it is obvious the cardinality of the union A ∪ B must equal the
sum of cardinalities of these disjoint sets.
The above statements assumed we were dealing with finite sets. What about infinite sets? Obviously the cardinality
of an infinite set must be infinite, but there is a major problem: not all infinities are the same! What do I mean by
that? Consider the integers Z. Clearly this set is infinite, more precisely, it’s countably infinite. Now consider R, the
set of reals. It is uncountably infinite. In some sense the set of reals is ’larger’ than the set of integers though both
8
are infinite. This can be seen by noting there is no natural mapping from the integers to the reals viz, 6 ∃ f : Z → R
such that f maps to every real number.
Cardinality as concept in set theory arose mainly through the work of Georg Cantor (1845 - 1918) who first defined
cardinality. Cantor realized the dilemma posed by infinite sets and devised a notation to denote different types
of infinities. The cardinality of countably infinite, well-ordered sets such as the integers, the natural numbers, the
rationals, the prime numbers and so forth is denoted by ℵ0 , pronounced ”aleph null”. Cantor also showed the reals
could be taken as the power set of the integers, so Card(R) = 2ℵ0 = ℵ1 , also known as c where the German fraktur c
represents ”continuum”. Every real number can be expressed by at least one infinite string of decimals. For example
1/2 may be written as 0.4999... . So every real number has a decimal part containing ℵ0 number of entries. Whence
the cardinality of the real numbers Card(R) must equal 2ℵ0 Cantor studied further generalizations on this theme. For
instance, the cardinality of the power set of real numbers P(R) equals ℵ2 = 2ℵ1 and so forth. Indeed ℵn+1 = 2ℵn . I
should mention ℵ0 differs conceptually from ∞; ℵn refers to the sizes of various infinite sets, whilst ∞ refers to the
extreme end of the number line. We’ll say more on this matter when we delve into topology.
Functions On Sets.
We now have acquired the necessary tools to discuss functions acting on sets. Given sets X and Y a function is a
single-valued mapping from X to Y . That is to say f : X → Y such that for x ∈ X ∃! y ∈ Y s.t f (x) = y. The
symbol ∃! employed here means ”exists uniquely”. The set X is called the domain of f and Y is referred to as the
codomain or range of f . Y is also know as the ” target space” of X by f . I should provide a cautionary note to the
student; some authors distinguish the codomain and range. The codomain is taken as the set of all possible values of
Y whilst the range is considered the actual set f (X). The two sets may differ. In that case the range f (X) is always
a subset of the codomain. For example, consider the function f : R → R defined by f (x) = x2 . The domain is R, the
range is {y ∈ R | y ≥ 0 } and the codomain is R. Usually one needn’t be too concerned about the difference between
codomain and range, but one should be aware some authors do distinguish them.
If a function f covers the entire target space Y , i.e., ∀ y ∈ Y , ∃ at least one x ∈ X s.t. y = f (x), the we say in
function is surjective or onto. In this case the range of f = ran(f ) = codomain(f ) = Y . If the function f is such
that ∀ y ∈ Y ∃ at most one x ∈ X such that y = f (x), then the function is said to be injective or one-to-one. This
is equivalent to saying ∀ x ∈ X, f (x1 ) 6= f (x2 ) whenever x1 6= x2 , or f (x1 ) = f (x2 ) =⇒ x1 = x2 . A function that
is both one-to-one and onto is said to be bijective or ”f is a bijection”. In that case, for each y ∈ Y there is exactly
one x ∈ X such that y = f (x).
Test yourself to see if you understand the concepts. Let X = Y = [ -1, 1 ]. Is f1 (x) = sin x injective? Is it surjective?
Is it bijective? Is f2 (x) = sin πx injective? Is it surjective? Is it bijective? Is f3 (x) =sin π2 x injective? Is is surjective?
Is it bijective? (Hint: one of these is only injective, one is only surjective, and only one is bijective).
Normally, we think of a function f (x) as merely a mathematical ”machine” that takes an ”input” x and produces
an output value y. That rather Victorian mechanistic viewpoint of functions, while certainly reasonable, does however
gloss over many subtle points. When we graph a function we’re visualizing it as a union of points, a curve if you
will, in R2 namely Graph(f )= {(x, y) | y = f (x)}. Since the modern view of mathematics is succintly encoded in
the catch-all phrase ”every mathematical object resides in some set” we should follow suit here as well. Henceforth
we shall consider the graph as the function itself ! That actually isn’t quite the radical departure one might suspect.
After all, we’re accustomed to graphing functions anyway, so stating the graph in a sense ”is” the function scarcely
qualifies as some monumental paradigm shift in our thinking. So with that in mind, we now state our new definition
of a function.
Definition of a Function:
A function f : X → Y is a set f ⊆ X x Y s.t. ∀ x ∈ X ∃! y ∈ Y s.t. (x, y) ∈ f .
We now posit functions as a single-valued mappings between sets. Functions we learned in junior high school such
as polynomials were typically assumed to have the entire real line i.e., R as their domains. We will however presently
pursue a slightly different course and look at functions between finite sets. This simpler setting permits a means
to introduce some necessary though abstract concepts required for topology. To begin, consider the figure below
9
illustrating a function, in this case f (x) = x2 , between two finite sets.
FIG. 6: The function f (x) = x2 .
The function in Figure 6 is easily delineated as a set viz, f = { (1, 1), (2, 4), (3, 9), (4, 16) }. Notice f in this
example is a bijection (1:1 and onto). A question naturally arises as to how many functions could we have created
between these sets. To answer that question, let us contemplate the issue of possible functions between two otherwise
arbitrary sets. Suppose we are given two sets X and Y with Card(X) = m, Card(Y ) = n. For each element x ∈ X
there are n possible choices for f (x). We conclude therefore the total number of possible functions must be nm . So
for the sets in Figure 6 there are 44 = 256 choices of functions. This leads us to the following definition:
Definition: The set of all possible functions f : X → Y is denoted by Y X .
The notation is intuitive; the number of functions is nm = Card(Y )Card(X) , whence the designation Y X for the
function space.
The above definition details the set of possible functions. One might rightfully ask how many such functions are
injective, or surjective, or bijective. The answers depend critically on the cardinalities of the two sets. In general X
and Y will have different cardinalities. That fact places constraints on the possible types of functions available. Let
f : X → Y . We have the following observations governing the behavior of f .
Proposition: Given f : X → Y , If Card(X) > Card(Y ) then f cannot be injective.
Proposition: Given f : X → Y , If Card(X) < Card(Y ) then f cannot be surjective.
Proposition: Given f : X → Y , f can be bijective only if Card(X) = Card(Y ).
These propositions are easily demonstrable and will prove quite useful when we cover topology. So let us consider
each one. If X has more elements than Y , then clearly some x-values will be mapped to the same y-value. Likewise,
if X possesses too few elements then it would be impossible to cover Y with f . Additionally, for each y to be uniquely
associated with a particular x, the available quantities of x’s and y’s must logically equate if all x-values are to be
used. This latter statement, that f maps every element x in the domain to some value in Y , was implicitly assumed
previously. We now consider it an explicit requirement.
Elements Of Point-Set Topology.
Now that elementary set theory has been covered, it is time to turn our attention to general topology also known
as point-set topology. There are a number of excellent books the motivated student should peruse. Classic works
such as Young and Hocking, Kelley, and Bourbaki come to mind though the student should read Bourbaki last as it’s
rather abstract. I also recommend Morris’ ”Topology Without Tears” - it is quite readable and ,better still, is freely
available as a PDF file (thank goodness for the internet!). So with that, let us begin our topological journey.
10
Basic Definitions And Axioms.
Let us first define exactly what is meant by a topology. Given a non-empty set X (X may be finite or infinite;
either will do for our purposes here), a topology on X, denoted by τX , is a subset of the power set of X (τX ⊆ P (X))
with the following properties:
• X ∈ τX .
• ∅ ∈ τX .
• X is taken as the universal set so X c = ∅ and ∅c = X.
• If U , V ∈ τX , then U ∪ V ∈ τX and U ∩ V ∈ τX .
• Element of τX are called open sets. So X and ∅ are open.
• Arbitrary unions (including possibly infinite unions) of open sets are open.
• Finite intersections of open sets are open.
• Complements of open sets are called closed sets. If U is open then U c := X/U is closed. Likewise the complement
of a closed is open.
These axioms define topologies. A set X taken together with its topology, i.e., (X, τX ), is called a topological space.
Often in the literature one reads something like ”Let X be a topological space...”, so it is important to remember that
authors tend toward brevity in their writings and that τX is assumed to be included (it’s just the unspoken part).
A couple of important points should be noted. First, I have presented the axioms for open sets, but one could just
as easily define a topology in terms of closed sets. Here are some of the above axioms modified for that case:
• X and ∅ are closed.
• Finite unions of closed sets are closed.
• Arbitrary (potentially infinite) intersections of closed sets are closed.
Secondly, the alert student has surely noticed X and ∅ are both open and closed! Such sets are called clopen sets,
which is a bad word-play on ”open” and ”closed”. The student may protest, ”How can a set be both open and closed?
That makes no sense!”. Granted, it’s not intuitive but the student must simply get used to it. After all, these are sets,
not refrigerator doors, so they may be simultaneously open and closed. For that matter, some subsets of X might be
neither open nor closed. Let us consider a simple example to clarify these concepts.
Example: Let X = { a, b, c, d } and let τX = { X, ∅, {a} , {a, b, d} , {b, d} }. Show that this is a topology and
classify all subsets. The set X as well as the null set are elements in τX , so those conditions are satisfied. The unions
and intersections of the other open sets are likewise contained within the topology, so this is a legitimate topology.
Now let’s classify all subsets of X:
• Open and closed (clopen) sets - X, ∅.
• Open but not closed sets - {a} , {a, b, d} , {b, d}.
• Closed and not open sets - {b, c, d} , {c} , {a, c}.
• Neither open nor closed sets - {b} , {d} , {a, b} , {a, d} , {b, c} , {c, d} , {a, b, c} , {a, c, d},