# Metric Spaces: Convergence. Open and Closed Sets

Definitions for open ball, center, radius, $$\epsilon$$​-neighbourhood, contact point, closure, closure operator.

Theorem 1

Closure has 4 properties:

1. If $$M \subseteq N$$ then $$[M] \subseteq [N]$$;

2. $$[[M]​] = [M]$$;

3. $$[M \cup N] = [M] \cup [N]$$;

4. $$[\emptyset] = \emptyset$$.

Definitions for limit point, isolated point, converge, limit.

Theorem 2

Necessary and sufficient condition for $$x$$ to be contact point of $$M$$ is the existence of a sequence $$\{x_n\}$$ of points of $$M$$ converging to $$x$$.

Theorem 2’

Same but with limit point.

Definitions of dense, everywhere dense, nowhere dense, separable, closed.

Theorem 3

Intersection of arbitrary number of closed sets is closed. Union of finite number of closed sets is closed.

Definition of interior point.

Theorem 4

Subset $$M$$ of metric space $$R$$ is open iff $$R-M$$ is closed.

Theorem 5

Union of arbitrary number of open sets is open. Intersection of finite number of open sets is open.

Theorem 6

Every open set $$G$$ on the real line is the union of a finite or countable system of pairwise disjoint open intervals.

## Problems

1. Give an example of a metric space $$R$$ and two open spheres $$S(x, r_1)$$ and $$S(y, r_2)$$ in $$R$$ such that $$S(x, r_1) \subseteq S(y, r_2)$$ although $$r_1 > r_2$$.

$$R$$ unit sphere. $$r_1 = 2$$, $$r_2 = 1$$.

2. Prove that every contact point of a set $$M$$ is either a limit point of $$M$$ or an isolated point of $$M$$.

Theorem 2 and definition.

3. Prove that if $$x_n \rightarrow x$$ and $$y_n \rightarrow y$$ as $$n \rightarrow \infty$$, then $$\rho(x_n, y_n) \rightarrow \rho(x, y)$$.

$$|\rho(x_n, x) = 0|$$, $$|\rho(y_n, y) = 0|$$, $$|\rho(x_n, y_n) - \rho(x, y)| \leq \rho(x_n, x) + \rho(y_n, y)|$$

4. $$f: X \rightarrow Y$$ mapping between metric spaces. Prove that $$f$$ is continuous at a point $$x_0$$ if and only if the sequence $$\{y_n\} = \{f(x_n)\}$$ converges to $$y = f(x_0)$$ whenever the sequence $$\{x_n\}$$ converges to $$x_0$$.

Show implication both ways.

5. Prove the statements:

1. The closure of any set $$M$$ is closed.

By definition.

2. $$[M]$$ is the smallest closed set containing $$M$$.

Trivial.

6. Is the union of infinitely many closed sets necessarily closed? How about the intersection of inifinitely many open sets? Give examples.

No.

7. Prove directly that the point $$\frac{1}{4}$$ belongs to the Cantor set $$F$$, although it is not an end point of any of the open intervals deleted in constructing $$F$$.

0, 2, 0, 2, …

8. $$F$$ Cantor set. Prove:

1. The points of the first kind $0, 1, \frac{1}{3}, \frac{2}{3}, \frac{1}{9}, \frac{2}{9}, \frac{7}{9}, \frac{8}{9}$ form an everywhere dense subset of $$F$$.

For any $$\epsilon$$ can find $$k$$ and $$n$$ s.t. x-ε < \frac{3k}{3^n} < x < \frac{3k+1}{3^n} < x + ε\$.

2. The numbers of the form $$t_1 + t_2$$, where $$t_1, t_2 \in F$$, fill the whole interval $$[0, 2]$$.

Jeffrey had the great idea of switching it into $$\frac{t_1+t_2}{2} \in [0, 1]$$. Then for each digit (i.e. 0, 1, 2), can choose for $$t_1$$ and $$t_2$$ to average out to it.

9. Given a metric space $$R$$, let $$A$$ be a subset of $$R$$ and $$x$$ a point of $$R$$. Then the number $\rho(A, x) = \inf_{a \in A} \rho(a, x)$ is called the distance between A and x. Prove that:

1. $$x \in A$$ imples $$\rho(A, x) = 0$$ but not conversely.

Easy.

2. $$\rho(A, x)$$ is a continuous function of $$x$$ (for fixed $$A$$).

Step through the steps.

3. $$\rho(A, x) = 0$$ if and only if $$x$$ is a contact point of A.

Easy.

4. $$[A] = A \cup M$$, where $$M$$ is the set of all points $$x$$ such that $$\rho(A, x) = 0$$.

Show implication both ways. Also use result from directly above.

10. Let $$A$$ and $$B$$ be two subsets of a metric space $$R$$. Then the number $\rho(A, B) = \inf_{a \in A; b \in B} \rho(a, b)$ is called the distance between A and B. Show that $$\rho(A, B) = 0$$ if $$A \cup B \neq \emptyset$$, but not conversely.

Easy enough.

11. Let $$M_K$$ be the set of all functions $$f$$ in $$C_{[a, b]}$$ satisfying a Lipschitz condition, i.e. the set of all $$f$$ such that $|f(t_1) - f(t_2)| \leq K|t_1 - t_2|$ for all $$t_1, t_2 \in [a, b]$$, where $$K$$ is a fixed positive number. Prove that

1. $$M_K$$ is closed and in fact is the closure of the set of all differentiable functions on $$[a, b]$$ such that $$|f \prime (t)| \leq K$$.

2. The set $M = \cup_K M_K$ of all functions satisfying a Lipschitz condition for some $$K$$ is not closed.

3. The closure of $$M$$ is the whole space $$C_{[a, b]}$$.

TODO solutions.

12. An open set $$G$$ in $$n$$​-dimensional Euclidean space $$R^n$$ is said to be connected if any points $$x, y \in G$$ can be joined by a polygonal line lying entirely in $$G$$. For example, the (open) disk $$x^2 + y^2 < 1$$ is connected, but not the union of the two disks $x^2 + y^2 < 1, \quad (x-2)^2 + y^2 < 1$ (even though they share a contact point). An open subset of an open set $$G$$ is called a component of $$G$$ if it is connected and is not contained in a larger connected subset of $$G$$. Use Zorn’s lemma to prove that every open set $$G$$ in $$R^n$$ is the union of no more than countably many pairwise disjoint components.

TODO solution.