Download Math 525 Notes for sec 22 Final Topologies Let Y be a set, {(X i,τi

Math 525 Notes for sec 22
Final Topologies
¯
¯
Let Y be a set, {(Xi , τi ) ¯ i ∈ I} a collection of topological spaces, and {gi ¯ i ∈ I} a corresponding family of functions, where gi : Xi → Y for all i ∈ I. Let µ be the topology on Y
defined by
U ∈ µ iff gi−1 (U ) ∈ τi for all i ∈ I.
Then µ is a topology on Y , and it is the finest topology on Y which makes all of the maps
gi : (Xi , τi ) → (Y, µ) continuous. The topology µ is called the final topology on Y induced
¯
¯
by {gi ¯ i ∈ I} and {(Xi , τi ) ¯ i ∈ I}.
Quotient Spaces
Suppose we let g : (X, τ ) → Y be a surjective map, and let µ be the final topology on Y
¯
induced by (X, τ ) and g. Then µ = {U ⊆ Y ¯ g −1 (U ) ∈ τ } is called the quotient topology
on Y , and (Y, µ) is called the quotient space of (X, τ ) relative to g. As discussed earlier,
the quotient topology µ is the finest topology on Y making g : (X, τ ) → (Y, µ) continuous.
To understand how “quotient spaces” work, consider a quotient (surjective) map g : (X, τ ) →
(Y, µ). Let ∼ be the equivalence relation defined on X defined by x ∼ y iff g(x) = g(y).
¯
The set {[x] ¯ x ∈ X} of equivalence classes relative to ∼ will be denoted by X̂. Note that
[x] = g −1 (g(x)).
Prop A38 In the notation of the preceding paragraph, σ : X̂ → Y defined by σ([x]) = g(x)
is a bijection. Let p : X → X̂ be the natural projection map p(x) = [x]. Then, in the
g
✲Y
diagram below, σ ◦ p = g.
X
p❘
✒
σ
X̂
Proof : For any x ∈ X, σ(p(x)) = σ([x]) = g(x), so σ ◦ p = g. To show σ is a bijection, note that σ is surjective because g is a quotient map (and therefore surjective), so
that for every y ∈ Y , we have y = g(x) = σ([x]) for some x ∈ X. σ is injective because
g(x1 ) = g(x2 ) ⇒ [x1 ] = [x2 ].
¥
In the preceding discussion, the map p is clearly surjective. Let µ̂ be the quotient topology
on X̂ induced by (X, τ ) and p. Then:
Prop A39 In Prop A38, σ : (X̂, µ̂) → (Y, µ) is a homeomorphism.
Proof : We already know from A38 that σ is a bijection. Now:
Claim 1: σ is continuous. To show this, let U ∈ µ. Then g −1 (U ) ∈ τ (since g : (X, τ ) →
(Y, µ) is continuous), and by A38, we have p−1 (σ −1 (U )) = g −1 (U ) ∈ τ . By definition of µ̂,
this means that σ −1 (U ) ∈ µ̂. Thus, σ is continuous.
Claim 2: σ −1 is continuous. To show this, let V ∈ µ̂. Then by definition of µ̂, p−1 (V ) ∈ τ .
but by A38, p−1 (V ) = g −1 (σ(V )). So σ(V ) ∈ µ, by definition of µ. Thus, σ −1 is continuous.
So σ is a continuous bijection whose inverse is continuous, so it is a homeomorphism.
¥
From a topological standpoint, two homeomorphic spaces are regarded as copies of the same
space. So we can, up to homeomorphism, consider every quotient space to be defined on a
quotient set. Considering (X̂, µ̂) as a quotient space of (X, τ ), it is interesting to note that
[
if A ⊆ X̂, then p−1 (A) =
{x}. Thus a set A is open in the quotient topology on X̂ iff
[x]∈A
[
{x} ∈ τ . Compare this with Munkres’ definition of a quotient map on page 137.
[x]∈A
A common way to construct quotient spaces is to start with a given space, and identify
certain points to form one or more equivalence classes, imposing the quotient topology on
the resulting quotient set.
Example 1 Let (X, τ ) be [0, 1] with its usual topology (that inherited as a subspace of
(R, τu )). Define a simple equivalence relation on [0, 1] by making 0 ∼ 1, and making all
other points equivalent only to themselves. Let a = [0] = [1] = {0, 1}. Then we can write
X̂ = (0, 1) ∪ {a}, where we identify any x ∈ (0, 1) with {x}. Then p : X → X̂ is defined by

 a if x = 0 or x = 1
p(x) =
.
 x if x ∈ (0, 1)
If x ∈ X̂ and x 6= a, then a basic nbhd of x in X̂ still has the form (x − ǫ, x + ǫ), where ǫ
is small enough so that (x − ǫ, x + ǫ) ⊆ (0, 1). A basic nbhd of a has the form W = U ∪ V ,
where U is a basic τ -nbhd of 0 and V is a basic τ -nbhd of 1. Note that p−1 (W ) looks like:
)
0
(
1
Since 0 and 1 are identified in X̂, we can visualize X̂ as a circle obtained by bending the line
segment [0, 1] into a circle and gluing the endpoints together. Topologically, this quotient
space is homeomorphic to a circle, considered as a subspace of R2 .
Example 2 Let (X, τ ) be the closed unit square [0, 1] × [0, 1], considered as a subspace of R2 .
We define a quotient set X̂ by identifying pairs of points on the two vertical boundaries which
are the same height above the x-axis (that is, identify (0, y) with (1, y) for all y ∈ [0, 1]). The
other points of the square are equivalent only to themselves. We can visualize X̂ as a cylinder
obtained by gluing together the vertical boundaries of X. When the quotient topology is
imposed on X̂, the resulting quotient space is homeomorphic to a cylinder embedded in R3 .
Example 3 Let (X, τ ) be as in Example 2, and define an equivalence relation by identifying
points diametrically opposed on the left and right vertical boundaries, That is, identify the
point (1, y) with the point (0, 1 − y) for all y ∈ [0, 1]. The resulting quotient space is homeomorphic to a Möbius strip.
Example 4 Define (X, τ ) as before, identify point (0, y) with point (1, y) for all y ∈ [0, 1],
and identify point (x, 0) with point (x, 1) for all x ∈ [0, 1]. The resulting quotient space is a
torus.
Example 5 For the same space (X, τ ), suppose boundary points are identified as follows:
(0, y) with (1, y) and (x, 1) with (1 − x, 0). The resulting quotient space is called a Klein
bottle.
None of the topological properties we’ve studied thus far are preserved by quotient maps, in
general.