Math Problems Vault

Problem 2.1.1

Prove Theorem 2.1.

Theorem 2.1: A set $E$ in a metric space $(M, d)$ is open if and only if its complement, $E^{c} = M \setminus E$ , is closed.

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Problem 2.1.2

Prove Theorem 2.2.

Theorem 2.2:

For any collection $\left\{ E_{\alpha }\right\}$ of open sets, $\bigcup_{\alpha } E_{\alpha }$ is open.
For any collection $\left\{ F_{\alpha }\right\}$ of closed sets, $\bigcap_{\alpha } F_{\alpha }$ is closed.
For any finite collection $\left\{ E_{i}\right\}_{i=1}^{n}$ of open sets, $\bigcap_{i=1}^{n} E_{i}$ is open.
For any finite collection $\left\{ F_{i}\right\}_{i=1}^{n}$ of closed sets, $\bigcup_{i=1}^{n} F_{i}$ is closed.

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Problem 2.1.3

Give examples to show that the word finite cannot be omitted in the last two parts of Theorem 2.2.

Theorem 2.2: (a) For any collection $\left\{ E_{\alpha }\right\}$ of open sets, $\bigcup_{\alpha } E_{\alpha }$ is open. (b) For any collection $\left\{ F_{\alpha }\right\}$ of closed sets, $\bigcap_{\alpha } F_{\alpha }$ is closed. (c) For any finite collection $\left\{ E_{i}\right\}_{i=1}^{n}$ of open sets, $\bigcap_{i=1}^{n} E_{i}$ is open. (d) For any finite collection $\left\{ F_{i}\right\}_{i=1}^{n}$ of closed sets, $\bigcup_{i=1}^{n} F_{i}$ is closed.

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Problem 2.1.4

Prove Theorem 2.3. Remember that your proof should be in a general metric space.

Theorem 2.3: If $E$ is a compact subset of a metric space, then $E$ is closed.

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Problem 2.1.5

(a)(b)

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(a)

Consider $\mathbb {R}$ with the Euclidean metric. Is $(0,1)$ open, closed, neither, or both? Explain. (It may surprise you to learn that a set can be neither open nor closed, or it can be both (a clopen set). In the rest of this exercise set you will meet examples of such sets.)

(b)

Consider $\mathbb {R}^{2}$ with the Euclidean metric. Is $(0,1)=\left\{ (x, 0) \mid 0<x<1\right\}$ open, closed, neither, or both? Explain.

Problem 2.1.6

Is it true that int $(A \cup B)=\operatorname {int}(A) \cup \operatorname {int}(B)$ ? Prove or give a counterexample. What if union is replaced by intersection?

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Problem 2.1.7

Let $M$ be any set with the discrete metric. What are the open sets?

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Problem 2.1.8

(a)(b)

Consider $\mathbb {R}$ with the Euclidean metric.

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(a)

Find the set of limit points of $\left\{ \frac{1}{n} \mid n=1,2,3, \ldots \right\}$ .

(b)

Find the set of limit points of $\mathbb {Q}$ .

Problem 2.1.9

Give an example of a closed and bounded set in a metric space that is not compact.

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Problem 2.1.10

Consider $\mathbb {R}$ with the Euclidean metric. Give an open cover of $(-10,10]$ that has no finite subcover.

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Problem 2.1.11

(a)(b)

This exercise is about the famous set due to Georg Cantor. If you do not know a definition of the Cantor set, look it up in a real analysis text or ask your professor for a reference.

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(a)

Is the Cantor set compact? Explain.

(b)

What is the interior of the Cantor set? Explain.

Problem 2.1.12

Prove that a totally bounded set is necessarily bounded. Then, using the discrete metric on any set of your choice, show that a bounded set need not be totally bounded.

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Problem 2.1.13

(a)(b)(c)

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(a)

Show that the closed unit ball in $C([0,1])$ is not compact by showing that it is not sequentially compact. That is, construct a sequence of functions with norm less than or equal to one that does not have a convergent subsequence.

(b)

Show that the closed unit ball in $\ell^{1}$ (respectively $\ell^{\infty }$ ) is not compact.

(c)

Show that the closed unit ball in any infinite-dimensional space is not compact.

Problem 2.1.14

Show that if $M$ is a metric space, $A \subseteq M$ is compact, and $U \subseteq M$ is open, then $A \backslash U$ is compact.

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Problem 2.1.15

(a)(b)

Consider metric spaces $M_{1}$ and $M_{2}$ .

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(a)

Prove that a function $f: M_{1} \rightarrow M_{2}$ is continuous if and only if the inverse image of every open set in $M_{2}$ is open in $M_{1}$ .

(b)

Assume that $A \subseteq M$ is compact and that $f: M \rightarrow \mathbb {R}$ is continuous. Show that $f(A)$ is a bounded subset of $\mathbb {R}$ .