Exercises

Exercises Exercises

1.

(a)

Determine the compact subsets of a topological space

X

with the indiscrete topology.

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(b)

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Determine the compact subsets of a topological space

X

with the indiscrete topology.

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2.

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Recall from Definition 12.13 that if

τ_{1}

and

τ_{2}

are two topologies on a set

X

such that

τ_{1} \subseteq τ_{2},

then

τ_{1}

is said to be a coarser (or weaker) topology than

τ_{2},

τ_{2}

is a finer (or stronger) topology than

τ_{1} .

In this exercise we explore the question of whether compactness is a property that is passed from weaker to stronger topologies or from stronger to weaker. Let

τ_{1}

and

τ_{2}

be two topologies on a set

X .

τ_{1} \subseteq τ_{2},

what does compactness under

τ_{1}

τ_{2}

imply, if anything, about compactness under the other topology? Justify your answers.

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3.

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Let

E

be the set of even integers, and let

τ = {Z} \cup {O \subseteq E} .

That is,

τ

is the collection of all subsets of

E

along with

Z .

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(a)

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Prove that

τ

is a topology on

Z .

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(b)

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Find all compact subsets of

(Z, τ) .

Verify your answer.

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(c)

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Prove or disprove: If

A

and

B

are compact subsets of a topological space

X,

then

A \cap B

is also a compact subset of

X .

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4.

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Let

(X, τ)

be a topological space

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(a)

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Prove that the union of any finite number of compact subsets of

X

is a compact subset of

X .

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(b)

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In Exercise 3 we should have seen that the intersection of compact sets is not necessarily compact. If

X

is Hausdorff, prove that the intersection of any finite number of compact subsets of

X

is a compact subset of

X .

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5.

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Consider

Z

with the digital line topology (see Exercise 11). Determine the compact subsets of

Z .

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6.

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For each

n \in Z^{+},

let

(- n, n)

be the set of integers in the interval

(- n, n)

(see Exercise 4.)

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(a)

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Show that

B = {(- n, n)}_{n \in Z^{+}}

is a basis for a topology

τ

Z

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(b)

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Is the subset

(- 2, 2)

compact in this topology?

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(c)

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Determine all of the compact subsets of

Z .

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7.

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Let

X

and

Y

be topological spaces, and let

f : X \to Y

be a function.

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(a)

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Suppose that

f

is a continuous function, and that

X

is compact and

Y

is Hausdorff. Prove that if

C

is a closed subset of

X,

then

f (C)

is a closed subset of

Y .

(Thus,

f

is a closed function.)

Hint.

Use Activity 17.3, Activity 17.4, and Theorem 17.6.

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(b)

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Suppose that

f

is a continuous bijection. Prove that if

X

is compact and

Y

is Hausdorff, then

f

is a homeomorphism.

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(c)

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Give an example where

f

is a continuous bijection and

X

is compact, but

f

is not a homeomorphism.

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(d)

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Give an example where

f

is a continuous bijection and

Y

is Hausdorff, but

f

is not a homeomorphism.

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8.

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The Either-Or topology on the interval

X = [- 1, 1]

has as its open sets all subsets of

X

that contain

(- 1, 1)

and any subset of

X

that doesn’t contain

0 .

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(a)

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Describe the non-trivial closed subsets of

X .

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(b)

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X

a Hausdorff topological space? Explain.

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(c)

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X

compact? Prove your answer.

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(d)

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Are there any subsets of

Z

that are not compact? Justify your answer.

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9.

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Let

K = {\frac{1}{k} ∣ k is a positive integer} .

Let

B

be the collection of all open intervals of the form

(a, b)

and all sets of the form

(a, b) ∖ K,

where

a < b

are real numbers as in Example 13.13. Let

τ_{K}

be the topology generated by

B .

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(a)

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Show that

(R, τ_{K})

is not compact.

Hint.

How is the

K

-topology related to the Euclidean topology?

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(b)

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Show that any subset of

R

that contains

K

is not a compact subset of

(R, τ_{K}) .

In particular, even though

[0, 1]

is a closed and bounded subset of

R

(R, τ_{K}),

we note that

[0, 1]

is not compact.

Hint.

Consider the sets

O_{k} = (\frac{1}{k}, 2) \cup (- 1, 1) ∖ K

for

k \in Z^{+} .

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10.

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Let

X

be a topological space.

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(a)

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Prove that if

X

is Hausdorff and

C

is a compact subset of

X,

then for each

x \in X ∖ C

there exist disjoint open sets

U

and

V

such that

x \in U

and

C \subseteq V .

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(b)

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Prove that if

X

is a compact Hausdorff space, then

X

is normal.

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11.

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Let

X

be a nonempty set and let

p

be a fixed element in

X .

Let

τ_{p}

be the particular point topology and

τ_{\overset{―}{p}}

the excluded point topology on

X .

That is

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$τ_{p}$ is the collection of subsets of $X$ consisting of $\emptyset,$ $X,$ and all of the subsets of $X$ that contain $p .$
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$τ_{\overset{―}{p}}$ is the collection of subsets of $X$ consisting of $\emptyset,$ $X,$ and all of the subsets of $X$ that do not contain $p .$

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That the particular point and excluded point topologies are topologies is the subject of Exercise 9 and Exercise 10. Determine, with proof, the compact subsets of

X

when

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(a)

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X

has the particular point topology

τ_{p}

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(b)

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X

has the excluded point topology

τ_{\overset{―}{p}} .

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12.

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In this exercise we encounter a non-Hausdorff topological space in which single points sets are closed, and in which compact subsets need not be closed. Consider the set

Z

with the finite complement topology

τ_{F C} .

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(a)

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Show that every single point set is closed.

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(b)

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Explain why

(Z, τ_{F C})

is not a Hausdorff space.

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(c)

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Let

U

be an open set in

(Z, τ_{F C})

that contains

0

and

V

an open set in

(Z, τ_{F C})

that contains

1 .

Explain why it cannot be the case that

U

and

V

are disjoint — that is,

U \cap V

must be non-empty.

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(d)

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Show that the subset

E

of even integers is a compact subset of

(Z, τ_{F C})

that is not closed. Verify your result.

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13.

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Let

(X, τ)

be a Hausdorff topological space.

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(a)

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Let

x \in X

and let

A

be a compact subset of

X .

Prove that there exist disjoint open subsets

U

and

V

X

such that

x \in U

and

A \subseteq V .

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(b)

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Let

A

and

B

be disjoint compact subsets of

X .

Prove that there exist disjoint open sets

U

and

V

such that

A \subseteq U

and

B \subseteq V .

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14.

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Let

(X, τ_{X})

be a topological space and let

A

be a subset of

X .

Let

τ_{A}

be the subspace topology on

A .

Prove that

A

is a compact subset of

X

if and only if

(A, τ_{A})

is a compact topological space.

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15.

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Let

X

be a topological space. A family

{F_{α}}_{α \in I}

of subsets of

X

is said to have the finite intersection property if for each finite subset

J

I,

⋂_{α \in J} F_{α} \neq \emptyset .

Prove that

X

is compact if and only if for each family

{F_{α}}_{α \in I}

of closed subsets of

X

that has the finite intersection property, we have

⋂_{α \in I} F_{α} \neq \emptyset .

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16.

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Even though

R

is not a compact space, if

x \in R,

then

x \in [x - 1, x + 1]

and so every point in

R

is contained in a compact subset of

R .

So if we view

R

from the perspective of a point in

R,

the space

R

looks compact around that point. This is the idea of local compactness. Locally compact spaces are important in the general topology of function spaces.

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Definition 17.15.

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A topological space

X

is locally compact if for each

x \in X

there is an open set

O

such that

p \in O

and

\overset{―}{O}

is compact.

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(a)

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Explain why

R^{n}

is locally compact for each

n \in Z^{+} .

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(b)

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Show that any compact space is locally compact.

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(c)

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Consider the Sorgenfrey line from Exercise 5. Recall that the Sorgenfrey line is the space

R

with a basis

B = {[a, b) ∣ a < b in R}

for its topology. Show that the Sorgenfrey line is Hausdorff but not locally compact.

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17.

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For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.

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