Math Problems Vault

Problem 1.4.1

A set $A$ of a metric space is said to be bounded if it is contained in some ball $B(x, r)$ . Show that a subset of a metric space which is compact must be bounded.

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Problem 1.4.2

Show that a finite union of compact sets is compact.

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Problem 1.4.3

(a)(b)(c)

Decide whether or not the following subsets of $\mathbb {R}$ are compact:

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(a)

$A = \left\{ 1/n : n \in \mathbb {N}\right\}$ ;

(b)

$B = \left\{ 0\right\} \cup A$ ;

(c)

$(0, 1]$ .

Problem 1.4.4

A topological space $X$ is called Hausdorff if given $x, y \in X$ there are disjoint open sets $U_x, U_y$ with $x \in U_x$ , $y \in U_y$ .

Show that in a Hausdorff space, compact sets are closed.

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Remark.

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In a general topological space, a set $U$ will be open if given $x \in U$ , then there is an open set $U(x)$ with $x \in U(x) \subset U$ ; for then we can write $U$ as the union of the sets $U(x)$ as $x$ ranges over the points of $U$ , and the union of open sets is open. With this criterion for a set to be open, the proof in the metric case can be modified to prove the result.

Problem 1.4.5

Consider the function $f : [0, 1) \to S^1$ given by $f(x) = (\cos 2\pi x, \sin 2\pi x)$ .

Show that $f$ is a bijection that is continuous but is not a homeomorphism.

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Remark.

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Consider the open set $(0, \tfrac {1}{2}) \subset [0, 1)$ and its image.

Problem 1.4.6

Analogous to the definition of least upper bound is that of greatest lower bound. Give a definition of greatest lower bound for a set $A \subset \mathbb {R}$ and use the least upper bound property to show that a set with a lower bound must have a greatest lower bound.

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Problem 1.4.7

Give an example of a closed, bounded subset $A$ of a metric space $X$ that is not compact.

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Remark.

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Consider the metric space $X$ itself to be a bounded noncompact subset of $\mathbb {R}$ and $A = X$ .