Basic properties of compact sets

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Problem 14.28 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 14.1 "Topological spaces"

(a)(b)(c)(d)

Let $X, Y$ be topological spaces, and let the set $M \subset X$ be compact. Prove that:

(a)

any closed subset of $M$ is compact;

(b)

if $X$ is Hausdorff, then $M$ is closed;

(c)

if $Y$ is Hausdorff, and the map $f: M \rightarrow Y$ is continuous, then $f(M)$ is compact in $Y$ ;

(d)

every infinite subset of $M$ has a limit point $x \in M$ .

Problem 14.30 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 14.1 "Topological spaces"

Let $M$ be a compact set in a topological space $X$ , and $f: M \rightarrow Y$ a continuous bijective map of $M$ onto a Hausdorff space $Y$ . Prove that the inverse map $f^{-1}$ is also continuous, i.e., $f$ is a homeomorphism of М and $Y$ .

Problem 14.34 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 14.2 "Linear topological spaces"

Suppose in a linear topological space the set $A$ is closed, and the set $B$ is compact. Is it true that the set $A+B$ is closed?

Problem 2.1.5 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.1 "Point Set Topology"

(a)(b)(c)

(a)

Define quotient map.

(b)

Show that if $X$ is compact, $Y$ is Hausdorff and $f : X \to Y$ is continuous and onto, then $f$ is a closed map.

(c)

Show that if $f$ satisfies the condition of (b) then $f$ is a quotient map.

Problem 2.1.9 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.1 "Point Set Topology"

Let $p : \tilde{X} \to X$ be an $n$ -sheeted covering projection, $n < \infty$ . Suppose that $X$ is compact. Prove that $\tilde{X}$ is compact.

Problem 1.4.1 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.4 "Compactness"

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.

Problem 1.4.2 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.4 "Compactness"

Show that a finite union of compact sets is compact.

Problem 1.4.3 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.4 "Compactness"

(a)(b)(c)

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

(a)

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

(b)

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

(c)

$(0, 1]$ .

Problem 1.4.4 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.4 "Compactness"

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.

Remark.

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 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.4 "Compactness"

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.

Remark.

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

Problem 1.4.7 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.4 "Compactness"

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

Remark.

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

Problem 1.5.5 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.5 "The product topology and compactness in $\reals[n]$"

Let $f : X \to Y$ be a continuous map of the compact metric space $(X, d_X)$ to the metric space $(Y, d_Y)$ . Show that $f$ is uniformly continuous.

Remark.

Use the Lebesgue number of the covering $\left\{ f^{-1}(B(y, \epsilon /2))\right\}_{y \in Y}$ of $X$ .

Problem 1.9.44 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

Consider the space $X = A \cup B$ , where $A = \left\{ (x, \sin 1/x) : 0 < x \leq 1\right\}$ and $B = \left\{ (0, y) : -1 \leq y \leq 1\right\}$ . The space $X$ is called the topologist's sine curve. This and the next two exercises show that $X$ is connected but not path connected.

The topologist’s sine curve—two views.

Show that $A \cup B = \overline{A}$ .

Problem 1.9.73 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

For each of the following subsets of $\mathbb {R}^{2}$ indicate which of the following properties it possesses, namely, (i) compact; (ii) connected; (iii) path connected; (iv) open; (v) closed:

(a)

$A = \left\{ (x_1, x_2) : x_1 \geq 0, 4 < x_2 \leq 8\right\}$ ;

(b)

$B = \left\{ (x_1, x_2) : x_1^2 + x_2^2 = 25\right\}$ ;

(c)

$C = A \cap B$ ;

(d)

$D = \left\{ (x_1, x_2) : x_1^2 + x_2^2 < 1\right\}$ ;

(e)

$E = \overline{D}$ .

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[Law06] Lawson T. Topology: A Geometric Approach. Oxford University Press, 2006.