Math 104
Topology
Quotient
spaces
Have you ever noticed that, when you write a fraction, the numerator and denominator do not go in
the same place? No! There is space between them.
But I digress. This handout is concerned with a different type of quotient space, as exemplified in the
following unorthodox-looking piece of arithmetic.
≈
Figure 1.
A topological quotient.
We shall make sense of this by the end of the handout.
The basic goal here is to make precise the notion of a quotient space. This had to happen eventually.
After all, “every surface is a quotient of a polygon,” whatever that means. The general strategy behind
dividing, or “modding,” out is well known, for instance, in algebra: one wishes to make something trivial,
hence one mods out by it. Indeed, the idea of quotient plays an ongoing role in all of serious mathematics.
Somewhat surprisingly, it also seems to appear in topology. As a result, in order to enhance our appreciation
for the discussion to follow as well as to furnish a nurturing environment in which our ideas may flourish,
we begin philosophically.
In mathematics, one likes to study objects called X.
Typically, X is endowed with some special structure. For instance, it might have a binary operation (or
two), a metric, a topology, and so on. For some reason, one might decide to replace X with a smaller yet
related object, probably to make it more tractable. Hence, one defines an equivalence relation ∼ on X and
studies the equivalence classes. For instance, in algebra, these classes often turn out to be cosets. In group
theory, they’re cosets of a subgroup and, in ring theory, cosets of a subring or ideal.
To fix some notation, given an element x in X, let [x] denote its equivalence class, that is,
[x] = {y ∈ X | x ∼ y}.
Then let X/∼ denote the set of all equivalence classes:
X/∼ = { [x] | x ∈ X }.
One hopes that studying the smaller set X/∼ will still shed some light back upon X, and therefore it is
natural to ask in advance that X/∼ retain as much of the structure of X as possible. A slick way of saying
this is to consider the function p: X → X/∼ that sends every element to its equivalence class, i.e., p(x) = [x],
and to demand that this function be “structure-preserving.” For example, in algebra, p: X → X/∼ should be
a homomorphism, and this imposes a real restriction on the types of equivalence relations that are allowed.
Thus in group theory, to define a quotient, one considers only cosets of normal subgroups, and, in ring theory,
cosets of two-sided ideals.
In topology, our X will be a topological space. From a geometric point of view, since typical “points”
of X/∼ are equivalence classes, we think of X/∼ as having been obtained from X by taking all the elements
of an equivalence class and collapsing them to a point, or gluing them all together. In keeping with the
philosophy outlined above, we declare that we are willing to call X/∼ a quotient space only if the following
conditions are met:
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(1) X/∼ should, like X, be a topological space; and
(2) p: X → X/∼ should be a continuous map.
Put more simply, we wish to topologize X/∼ in a way satisfying condition (2). There seems to be no good
reason to place any further conditions on what a quotient space should be, so, with this motivation, we make
the following definition.
Definition. Suppose that X is a topological space on which an equivalence relation ∼ has been defined.
Let X/∼ denote the set of equivalence classes, and let p : X → X/∼ be defined by p(x) = [x]. Then define
a subset U ⊂ X/∼ to be open if p−1 (U ) is an open set in X. The collection of all open sets thus defined is
called the quotient topology on X/∼. When equipped with this topology, X/∼ is called a quotient space of
X.
Of course, one must verify that these open sets really do satisfy the axioms for being a topology on
X/∼, a straightforward exercise. That being done, it is obvious from the definition that p : X → X/∼
is continuous. Indeed, the choice of which subsets of X/ ∼ to call open was made just so to meet this
requirement. Thus, conditions (1) and (2) have been met. But there’s even more to it—namely, the quotient
topology is completely determined by p. That is to say, if one wishes to know whether a given subset
A ⊂ X/∼ is open, it is a matter of definition that one should just check whether p−1 (A) is open in X. (In
contrast, for arbitrary continuous maps f : Y → Z, it is not generally true that f −1 (A) open in Y =⇒ A
open in Z.)
Note that the preceding construction works for any equivalence relation on X, so, unlike the situation
in algebra, there is no need to draw a distinction between those relations that are quotient-able and those
that are not. Such refinement of taste is not a prerequisite for topology.
Witness the ever-popular Parade of Doughnuts:
etc.
torus
2-holed torus
Figure 2.
3-holed torus
The Parade of Doughnuts.
Before moving on to examples, a few more words on notation. First, as a matter of exposition, the
equivalence relation used to produce a given quotient is usually not described in complete detail. For
instance, x ∼ x in any equivalence relation, so this is not mentioned usually. Similarly, x ∼ y if and only if
y ∼ x; it is not customary to list both. Finally, we usually also omit equivalences that follow by transitivity
from those equivalences already listed.
The examples we consider shall frequently involve the following spaces:
(i) I = [0, 1] = {x ∈ R | 0 ≤ x ≤ 1}, the unit interval;
(ii) B n = {x ∈ Rn | kxk ≤ 1}, the closed unit n-ball;
(iii) S n = {x ∈ Rn+1 | kxk = 1}, the unit n-sphere.
Note that S n ⊂ B n+1 and that, as a special case of this, S 0 = {−1, 1} ⊂ [−1, 1] = B 1 .
As a first example, let X be the unit square I × I, and specify an equivalence relation on X by defining
x×0∼x×1
0 × y ∼ 1 × (1 − y)
and
The identifications are illustrated in Figure 3, and they should look familiar.
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(x, 1)
a
(1, 1-y)
or
b
b
(0, y)
a
(x, 0)
Figure 3.
A familiar quotient space.
This shows that the thing we have been calling a Klein bottle really does exist as a topological space.
Just to get some feel for its topology, let us consider the two subsets of X/∼ given by
U = { [x × y] | 1/2 < x < 3/4 }
V = { [x × y] | 1/2 < y < 3/4 }
and
and ask whether they are open in the quotient topology. (Note that the elements of U and V are equivalence
classes, as they should be.) Proceeding according to the definition, we examine the pre-image under p of
each set in I × I. These are shown in Figure 4. Clearly, p−1 (U ) is open in I × I, while p−1 (V ) is not. Thus
U is an open set in the Klein bottle, and V isn’t.
p
-1 (U)
p -1 (V) =
=
p−1 (U ) and p−1 (V ).
Figure 4.
∗
∗
∗
We now run through some of the standard selections of the topological repertoire.
I. Wedges.
Let X and Y be two spaces, and choose one point x0 and y0 , respectively, from each. On the union
X ∪ Y , define an equivalence relation by just setting x0 ∼ y0 . To envision the resulting quotient (X ∪ Y )/∼,
one should simply take X and Y and attach x0 to y0 , i.e., glue the spaces together at a point. The result is
called the wedge of X and Y relative to the base points x0 and y0 and is denoted X ∨ Y . Pictured in Figure
5 are S 1 ∨ S 1 and S 1 ∨ S 2 .
S1 ∨ S1
S1 ∨ S2
Figure 5.
Two wedges.
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In truth, all this is somewhat unnecessary since X ∨Y is homeomorphic to the subspace (X ×y0 )∪(x0 ×Y )
inside X × Y . That is, the subspaces X × y0 and x0 × Y are copies of X and Y within X × Y that are
already attached at a single point, namely, at the point x0 × y0 . Thus, it is possible to define X ∨ Y without
resorting to any kind of quotient argument. Nonetheless, if one really wishes to glue together two spaces at
a point, it seems more natural to visualize it in the way we described first.
II. Collapsing subspaces.
Suppose that X is a space and that A ⊂ X. Define an equivalence relation on X by setting a ∼ b for
all elements a, b in A. In forming the quotient, all points in A are collapsed to a single point, while points
in X − A are left alone. The resulting space is denoted X/A. For instance, B 2 /S 1 is homeomorphic to the
2-sphere S 2 , as seen in Figure 6.
B2 =
begin
collapsing
S1
Figure 6.
finish
collapsing
= S2
Collapsing S 1 ⊂ B 2 .
Some particularly important examples of collapsing are the following.
A. Cones.
Given a space X, consider the “cylinder” X × I.
Xµ1
Xµ I =
X=
X µ0
Figure 7.
The cylinder on X.
If the subset X × 1 is now collapsed to a point, the result is called the cone on X.
Definition. The cone on X, denoted C(X), is defined by
C(X) = (X × I)/(X × 1).
(See Figure 8.) For instance, C(S 1 ) is homeomorphic to B 2 , and, more generally, C(S n−1 ) = B n .
B. Suspensions.
What happens if you now collapse the other end of the cylinder, too?
Definition. The suspension of X, denoted S(X), is defined by
S(X) = C(X)/(X × 0).
(See Figure 9.) For example, S(S 1 ) = S 2 , and, more generally, S(S n−1 ) = S n .
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formerly X µ 1
C(X) =
X µ (1/2)
S(X) =
Xµ0
formerly X µ 0
Figure 8.
The cone on X, C(X).
Figure 9.
The suspension of X, S(X).
It is often convenient to think of the suspension S(X) as the union of two cones attached to one another
along their bases. In Figure 9, these cones correspond to the subsets
C1 = { [x × t] | x ∈ X, 0 ≤ t ≤ 1/2 }
and C2 = { [x × t] | x ∈ X, 1/2 ≤ t ≤ 1 },
each having X × (1/2) as base.
C. Smash products.
We observed earlier that the wedge X ∨ Y may be regarded as a subset of X × Y . Let’s smash it to a
point.
Definition. The smash product of X and Y , denoted X ∧ Y , is defined by
X ∧ Y = (X × Y )/(X ∨ Y ).
Though we shall not have much use for the smash product in this seminar, it is a useful thing to know
about later in life. For instance, the smash product of two spheres is again a sphere: S n ∧ S m = S n+m . In
the problem set, you are asked to verify this in the case n = m = 1. In fact, this is the quotient pictured on
the first page of this handout.
As a much more pedestrian example, let’s take the smash product I ∧ I. In the definition of I ∨ I, we
choose x0 = y0 = 0 as base point in each interval and proceed as shown in Figure 10. It’s clear from the
picture that I ∧ I is homeomorphic to B 2 and, indeed, to I × I itself (so collapsing the wedge had little
effect).
collapse I ∨ I =
(I × 0) ∪ (0 × I)
I ×I =
=I ∧I
Figure 10.
The smash product I ∧ I.
III. Attaching cells
Lastly, we consider a particularly flexible method of getting at many useful spaces. The idea is to
construct the space inductively, building it up one dimension at a time.
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The initial input for the construction is a topological space X and a continuous map f : S n−1 → X.
Keep in mind that S n−1 ⊂ B n . We wish to attach the closed ball B n to X by identifying every point z
in S n−1 with its image f (z) in X. More formally, on the union X ∪ B n , define an equivalence relation by
setting f (z) ∼ z for all z in S n−1 . The resulting quotient, sometimes called an adjunction space, is denoted
X ∪f en :
X ∪f en = (X ∪ B n )/ ∼ .
One says that “an n-cell has been attached to X.” Figure 11 shows the situation in the case of low-dimensional
cells.
(a) Attaching a 1-cell:
f(S0)
f
-1
1
S0
X ∪f e1
X
(b) Attaching a 2-cell:
f(S1)
f
S1
X ∪f e2
X
Figure 11.
Attaching low-dimensional cells.
As a concrete example, suppose that we wish to build a 2-sphere S 2 using these ideas. One way to do
it is to start with two points p and q and attach first two 1-cells and then two 2-cells, as indicated in the
sequence shown in Figure 12. Four attaching maps f, g: S 0 → (p ∪ q) and h, k: S 1 → (p ∪ q ∪ (e1 )1 ∪ (e1 )2 )
are used in this construction, each looking rather like an appropriate identity map.
(e 2 ) 1
(e 1 ) 1
-1
1
attach
two
1-cells
p
q
(e 1 ) 2
attach
two
2-cells
p
q
(e 2 ) 2
Figure 12.
S 2 = p ∪ q ∪ (e1 )1 ∪ (e1 )2 ∪ (e2 )1 ∪ (e2 )2 .
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e2
f
p
S1
p
S 2 = p ∪ e2 .
Figure 13.
Alternatively, one could construct S 2 more efficiently by starting with a single point p and immediately
attaching a 2-cell via the constant map f : S 1 → p. This is illustrated in Figure 13.
Thus a single space can be built up in more than one way. Indeed, given a space X, one might try to
study the different ways in which it may be decomposed into cells. The actual number of cells of various
dimensions is likely to vary, but, just from the example of S 2 , you can probably guess an interesting invariant
that is lurking in the background.
That just about does it for now. In parting, we should acknowledge that one could also develop quotient
spaces through the concept of “quotient map.” This seemingly more general approach does not really yield
anything new, however: up to homeomorphism, every quotient space is of the form X/∼ for some suitably
chosen equivalence relation on X (Munkres, Corollary 22.3(a)).
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