Quotient Spaces

Section Quotient Spaces

As we saw in our preview activity, if we have a surjection

p

from a topological space

(X, τ)

to a set

Y,

we were able to define a topology on

Y

by making the open sets the sets

U \subseteq Y

such that

p^{- 1} (U)

is open in

X .

This is how we will create what is called the quotient topology. Before we can define the quotient topology, we need to know that this construction always makes a topology.

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Activity 16.2.

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Let

(X, τ_{X})

be a topological space, let

Y

be a set, and let

p : X \to Y

be a surjection. Let

τ_{Y} = {U \subseteq Y ∣ p^{- 1} (U) \in τ_{X}} .

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(a)

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Why are

\emptyset

and

Y

τ_{Y} ?

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(b)

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Let

{U_{β}}

be a collection of sets in

τ_{Y}

for

β

in some indexing set

J .

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(i)

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Show that

⋃_{β \in J} U_{β}

is in

τ_{Y} .

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(ii)

🔗

J

is finite, show that

⋂_{β \in J} U_{β}

is in

τ_{Y} .

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(c)

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What conclusion can we draw about

τ_{Y} ?

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Activity 16.2 allows us to define the quotient topology.

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Definition 16.1.

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Let

(X, τ_{X})

be a topological space, let

Y

be a set, and let

p : X \to Y

be a surjection.

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The quotient topology on $Y$ is the set

${U \subseteq Y ∣ p^{- 1} (U) \in τ_{X}} .$
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Any function $p : X \to Y$ is a quotient map if $p$ is surjective and for $U \subseteq Y,$ $U$ is open in $Y$ if and only if $p^{- 1} (U)$ is open in $X .$
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If $p : X \to Y$ is a quotient map, then the space $Y$ is the quotient space of $X$ determined by $p .$

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Activity 16.3.

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(a)

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Let

X = R

with standard topology, let

Y = {- 1, 0, 1},

and define

p : X \to Y

p (x) = {\begin{cases} 1 & if x > 0 \\ 0 & if x = 0 \\ - 1 & if x < 0. \end{cases}

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Find all of the open sets in the quotient topology.

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(b)

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Let

X = R

with standard topology, let

Y = [0, 1),

and define

p : X \to Y

p (x) = x - ⌊ x ⌋,

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where

⌊ x ⌋

is the largest integer less than or equal to

x,

(For example

⌊ 1.2 ⌋ = 1,

and so

p (1.2) = 1.2 - 1 = 0.2 .

The function defined by

⌊ x ⌋

is also called the floor function. Be careful, note that

⌊ - 0.7 ⌋ = - 1 .

) Determine the sets in the quotient topology.

Hint.

The graph of

p

[- 2, 2]

is shown in Figure 16.2.

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Another perspective of the quotient topology utilizes the fact that any equivalence relation

\sim

on a set

X

partitions

X

into a union of disjoint equivalence classes

[x] = {y \in X ∣ y \sim x} .

There is a natural surjection

q

from

X

to the space of equivalence classes given by

q (x) = [x] .

We investigate this perspective in the next activity.

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Activity 16.4.

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Let

X = {a, b, c, d, e, f}

and let

τ = {\emptyset, {a}, {b}, {a, b}, {a, b, c}, {a, b, c, d}, X} .

Then

(X, τ)

is a topological space. Let

A = {a, b, c}

and

B = {d, e, f} .

Define a relation

\sim

X

such that

x \sim y

x

and

y

are both in

A

or both in

B .

Assume that

\sim

is an equivalence relation. The sets

A

and

B

are the equivalence classes for this relation. That is

A = [a] = [b] = [c]

and

B = [d] = [e] = [f] .

Let

X^{*} = {A, B} .

Then we can define

p : X \to X^{*}

by sending

x \in X

to the set to which it belongs. That is,

p (x) = [x]

for

x \in X,

p (a) = A, p (b) = A, p (c) = A, p (d) = B, p (e) = B, and p (f) = B .

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Determine the sets in the quotient topology on

X^{*} .

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The partition of

X

in Activity 16.4 into the disjoint union of sets

A

and

B

defines an equivalence relation on

X

where

x \sim y

x

and

y

are both in the same set

A

B .

That is,

a \sim b \sim c

and

d \sim e \sim f .

In this context, the sets

A

and

B

are equivalence classes —

A = [a]

and

B = [d],

where

[x]

is the equivalence class of

x .

This leads to a general construction.

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(X, τ)

is a topological space and

\sim

is an equivalence relation on

X,

we can let

X / \sim

be the set of distinct equivalence classes of

X

under

\sim .

Then

p : X \to X / \sim

defined by

p (x) = [x]

is a surjection and

X / \sim

has the quotient topology. The space

X / \sim

is called a quotient space. The space

X / \sim

is also called an identification space because the equivalence relation identifies points in the set to be thought of as the same. This allows us to visualize quotient spaces as resulting from gluing or collapsing parts of the space

X .

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Figure 16.3. A tube as the identification space $X / \sim .$

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Example 16.4.

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Let

I = [0, 1]

and let

X = I \times I

with standard topology. Define a relation

\sim

X

(x, y) \sim (x, y)

0 < y < 1

and

0 \leq x \leq 1,

(x, 0) \sim (x, 1)

0 \leq x \leq 1 .

It is straightforward to show that

\sim

is an equivalence relation. Let us consider what the identification space

X / \sim

looks like. The space

I \times I

is the unit square as shown in Figure 16.3. All points in the interior of the square are identified only with themselves. However, the top side and bottom side are identified with each other in the same direction. Think of

X

as a piece of paper. We roll up the sides of the square to make the top and bottom sides coincide. The result is that

X / \sim

is the cylinder as shown in Figure 16.3.

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Activity 16.5.

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Quotient spaces can be difficult to describe. This activity presents a few more examples.

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(a)

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Let

X = [0, 1]

with standard topology and define an equivalence relation

\sim

X

0 \sim 1

and

x \sim x

for all

x

not equal to

0

1 .

What does the quotient space

X / \sim

look like?

Figure 16.5. From left to right: the identifications for parts (i), (ii), and (iii)

Hint.

Think about the relation

\sim

as gluing the points

0

and

1

together.

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(b)

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Describe quotient spaces of

X = I \times I

with standard topology given by the following equivalence relations

\sim .

Depictions of the identifications are shown in Figure 16.5. (Here

I

is the closed interval

[0, 1] .

)

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(i)

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(x, y) \sim (x, y)

0 < y < 1

and

0 \leq x \leq 1

and

(x, 0) \sim (1 - x, 0)

when

0 \leq x \leq 1 .

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(ii)

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(x, y) \sim (x, y)

0 < x < 1

and

0 < y < 1,

(x, 0) \sim (x, 1)

for

0 < x < 1,

(0, y) \sim (1, y)

for

0 < y < 1,

and

(0, 0) \sim (0, 1) \sim (1, 0) \sim (1, 1) .

(This space is called a Möbius strip.)

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(iii)

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(x, y) \sim (x, y)

0 < x < 1

and

0 < y < 1

and

(x, y) \sim (u, v)

(x, y)

and

(u, v)

are boundary points.

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Many other interesting identification spaces can be made. For example, let

X = I \times I

and define

\sim

(x, y) \sim (x, y)

0 < x < 1

and

0 < y < 1,

(0, y) \sim (1, y)

for

0 < y < 1,

(x, 0) \sim (1 - x, 1)

for

0 < x < 1 .

This identification is illustrated in Figure 16.6. The resulting identification space

X / \sim

is a Klein bottle. A nice illustration of this can be seen at maths.org ⁸.

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Figure 16.6. Identifications for the Klein Bottle.

plus.maths.org/content/introducing-klein-bottle

Topology: An Inquiry-Based Approach

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Section Quotient Spaces

Activity 16.2.

(a)

(b)

(i)

(ii)

(c)

Definition 16.1.

Activity 16.3.

(a)

(b)

Activity 16.4.

Example 16.4.

Activity 16.5.

(a)

(b)

(i)

(ii)

(iii)