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. \(r_1 = 2\), \(r_2 = 1\).

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.

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

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

\(f: X \rightarrow Y\) 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.

Prove the statements:

1. The closure of any set \(M\) is closed.

   By definition.

2. \([M]\) is the smallest closed set containing \(M\).

   Trivial.

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

No.

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, …

\(F\) 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.

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.

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.

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.

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.