Exercises

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Exercises Exercises

1.

(a)

Let

(Y_{1}, τ_{1})

and

(Y_{2}, τ_{2})

be topological spaces, where

Y_{1} = {a, b, c}

with

τ_{1} = {\emptyset, {a}, {b, c}, Y_{1}}

and

Y_{2} = {1, 2}

with

τ_{2} = {\emptyset, {1}, Y_{2}} .

Find all of the sets of the form

S = {π_{1}^{- 1} (O_{1}) ∣ O_{1} is open in Y_{1}} \cup {π_{2}^{- 1} (O_{2}) ∣ O_{2} is open in Y_{2}}

and verify that these sets generate the product topology on

Y_{1} \times Y_{2} .

(b)

Let

X_{i}

for

i

from

1

to

n

be topological spaces and let

X = Π_{i = 1}^{n} X_{i} .

Show that the collection

S = ⋃_{i = 1}^{n} {π_{i}^{- 1} (O_{i}) ∣ O_{i} is open in X_{i}}

is a subbasis for the box topology on

X .

2.

Let

X

be the set of real numbers with the standard Euclidean metric topology and let

Y

be the real numbers with the discrete topology.

(a)

Explain why the set of all “horizontal intervals” of the form

I = (a, b) \times {c} = {(x, c) ∣ a < x < b}

is a base for the product topology on

X \times Y .

(b)

Find the interior and closure of each of the following subsets of

X \times Y .

(i)

A = {(x, 0) ∣ 0 \leq x < 1}

(ii)

B = {(0, y) ∣ 0 \leq y < 1}

(iii)

C = {(x, y) ∣ 0 \leq x < 1, 0 \leq y < 1}

3.

Let

X_{1}

and

X_{2}

be topological space and let

A_{1}

be a subset of

X_{1}

and

A_{2}

a subset of

X_{2} .

Assume the product topology on

X_{1} \times X_{2} .

Prove each of the following.

(a)

\overset{―}{A_{1} \times A_{2}} = \overset{―}{A_{1}} \times \overset{―}{A_{2}}

(b)

Int (A_{1} \times A_{2}) \subseteq Int A_{1} \times Int A_{2}

4.

Let

X

and

Y

be topological spaces and let

{U_{α}}_{α \in I}

and

{V_{β}}_{β \in J}

be collections of open sets in

X

and

Y,

respectively, for some indexing sets

I

and

J .

Show that

⋃_{\begin{matrix} α \in I \\ β \in J \end{matrix}} (U_{α} \times V_{β}) = (⋃_{α \in I} U_{α}) \times (⋃_{β \in J} V_{β}) .

5.

(a)

If

S_{1},

S_{2},

T_{1},

and

T_{2}

are sets, show that

(S_{1} \times T_{1}) \cap (S_{2} \times T_{2}) = (S_{1} \cap S_{2}) \times (T_{1} \cap T_{2}) .

(b)

If

B_{X}

is a base for a topology

τ_{X}

on a space

X

and

B_{Y}

is a base for a topology

τ_{Y}

on a space

Y,

show that

B_{X} \times B_{Y}

is a base for the product topology on

X \times Y .

6.

Prove that the product of any finite number of connected spaces is connected.

7.

Prove that the product of any finite number of compact spaces is compact.

8.

(a)

Prove that if

(X_{1}, τ_{1})

and

(X_{2}, τ_{2})

are path connected topological spaces, then

X_{1} \times X_{2}

with the product topology is a path connected topological space.

(b)

Prove that the product of any finite number of path connected spaces is path connected.

9.

Let

X_{1}

and

X_{2}

be topological spaces and

X = X_{1} \times X_{2} .

Is it true that if

X

is compact, then both

X_{1}

and

X_{2}

are compact. Prove your answer.

10.

Let

X_{1}

and

X_{2}

be topological spaces and let

π_{i} : X_{1} \times X_{2} \to X_{i}

be the projection mapping. We have shown that

π_{i}

is continuous. Now show that

π_{i}

is an open map for

i

equal 1 and 2. Assume the standard product topology.

11.

Let

X_{1}

and

X_{2}

be topological spaces with cardinalities at least 2. Let

X = X_{1} \times X_{2} .

Prove that the product space topology on

X = X_{1} \times X_{2}

is the discrete topology if and only if the topologies on

X_{1}

and

X_{2}

are the discrete topologies.

12.

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. Throughout, let

X_{1}

and

X_{2}

be topological spaces and

X = X_{1} \times X_{2}

with the product topology.

(a)

If

A_{1} \subseteq X_{1}

and

A_{2} \subseteq X_{2},

then

(X_{1} \times X_{2}) ∖ (A_{1} \times A_{2}) = (X_{1} ∖ A_{1}) \times (X_{2} ∖ A_{2}) .

(b)

If

A_{1} \subseteq X_{1}

and

A_{2} \subseteq X_{2},

then

Bdry (A_{1} \times A_{2}) \subseteq Bdry A_{1} \times Bdry A_{2} .

(c)

If

A_{1} \subseteq X_{1}

and

A_{2} \subseteq X_{2},

then

Bdry A_{1} \times Bdry A_{2} \subseteq Bdry (A_{1} \times A_{2}) .

(d)

If

O_{1}

is an open subset of

X_{1}

and

O_{2}

is an open subset of

X_{2},

then

O_{1} \times O_{2}

is an open subset of

X_{1} \times X_{2} .

(e)

If

O_{1}

is a subset of

X_{1}

and

O_{2}

is a subset of

X_{2}

and

O_{1} \times O_{2}

is an open subset of

X_{1} \times X_{2},

then

O_{1}

is an open subset of

X_{1}

and

O_{2}

is an open subset of

X_{2} .

(f)

If

C_{1}

is a closed subset of

X_{1}

and

C_{2}

is a closed subset of

X_{2},

then

C_{1} \times C_{2}

is a closed subset of

X_{1} \times X_{2} .

(g)

If

C_{1}

is a subset of

X_{1}

and

C_{2}

is a subset of

X_{2}

and

C_{1} \times C_{2}

is a closed subset of

X_{1} \times X_{2},

then

C_{1}

is a closed subset of

X_{1}

and

C_{2}

is a closed subset of

X_{2} .

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