Chapter 6

Prove that every contact point of a set \(M\) is either a limit point of \(M\) or an isolated point of \(M\). Comment. In particular, \([M]\) can only contain points of the following three types: a) Limit points of \(\mathrm{M}\) belonging to \(\mathrm{M}\); b) Limit points of \(\mathrm{M}\) which do not belong to \(\mathrm{M}\); c) Isolated points of \(\mathrm{M}\). Thus \([M]\) is the union of \(M\) and the set of all its limit points.

Prove that a) The closure of any set \(M\) is a closed set; b) \([M]\) is the smallest closed set containing M.

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

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 \(\mathrm{F}\). Hint. The point \(\frac{1}{4}\) divides the interval \([0,1]\) in the ratio \(1: 3\). It also divides the interval \(\left[0, \frac{1}{3}\right]\) left after deleting \(\left(\frac{1}{3}, \frac{2}{3}\right)\) in the ratio \(3: 1\), and so on.

Let \(A\) and B be two subsets of a metric space \(R\). Then the number $$ \rho(A, \mathrm{~B})=\inf _{a \in A \atop b \in B} \rho(a, b) $$ is called the distance between \(\mathrm{A}\) and \(\mathrm{B}\). Show that \(\rho(A, \mathrm{~B})=0\) if A \(\mathrm{n} \mathrm{B} \neq \varnothing\), but not conversely.

Let \(M_{K}\) be the set of all functions \(\mathbf{f}\) in \(C_{[a, b]}\) satisfying a Lipschitz condition, i.e., the set of allf such that $$ \left|f\left(t_{1}\right)-f\left(t_{2}\right)\right| \leqslant K\left|t_{1}-t_{2}\right| $$ for all \(t_{1}, t_{2} \in[a, b]\), where \(K\) is a fixed positive number. Prove that a) \(M_{K}\) is closed and in fact is the closure of the set of all differentiable functions on \([a, b]\) such that \(\left|f^{\prime}(t)\right| \leqslant K\); b) The set $$ M=\bigcup_{K} M_{K} $$ of all functions satisfying a Lipschitz condition for some \(K\) is not closed; c) The closure of \(M\) is the whole space \(C_{[a, b]}\).

An open set G in n-dimensional Euclidean space \(\mathrm{R}^{\mathrm{n}}\) is said to be connected if any points \(x, y \in G\) can be joined by a polygonal line \(^{B}\) lying entirely in G. For example, the (open) disk \(\mathrm{x}^{2}+\mathrm{y}^{2}<1\) 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 \(\mathrm{G}\) is called a component of \(\mathrm{G}\) if it is connected and is not contained in a larger connected subset of \(\mathrm{G}\). Use Zorn's lemma to prove that every open set \(\mathrm{G}\) in \(\mathrm{R}^{\mathrm{n}}\) is the union of no more than countably many pairwise disjoint components. Comment. In the case \(\mathrm{n}=1\) (i.e., on the real line) every connected open set is an open interval, possibility one of the infinite intervals \((-\infty, c o)\), \((a, \infty),(-\infty, b)\). Thus Theorem 6 on the structure of open sets on the line is tantamount to two assertions: 1) Every open set on the line is the union of a finite or countable number of components; 2) Every open connected set on the line is an open interval. The first assertion holds for open sets in \(R^{n}\) (and in fact is susceptible to further generalizations), while the second assertion pertains specifically to the real line.