\documentclass[12pt]{article} \usepackage{theorem,amsmath,amssymb} \input{listki.tex} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{4}{GEOMETRY 4: Topology of metric spaces.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{opredelenie} Let $M$ be a metric space and $X\subseteq M$. Then $X$ is called {\bf open} when it contains, together with any point $x\in X$, some $\epsilon$-ball with the center in $x$. A subset is called {\bf closed} if its complement is open. \end{opredelenie} \begin{zadacha} Prove that $X$ is open iff for any sequence $\{a_i\}$ converging to $x\in X$ all but a finite number of $a_i$ belong to $X$. \end{zadacha} \begin{zadacha} Prove that the union of any number of open sets is open. Prove that the intersection of a finite number of closed sets is closed. \end{zadacha} \begin{zadacha} Prove that the closed ball $$ \overline B_\epsilon(x) = \{ y \in X \ \ | \ \ d(x,y)\leq \epsilon\} $$ is a closed subset. \end{zadacha} \begin{zadacha} Prove that a set is closed iff it contains all its accumulation points. \end{zadacha} \begin{opredelenie} The {\bf closure} of a set $A\subset M$ is the union of $A$ and the set of all the accumulation points of $A$. \end{opredelenie} \begin{zadacha} Consider a metric space, a closed ball $\overline B_\epsilon(x)$ and an open ball $B_\epsilon(x)$. Is it always true that $\overline B_\epsilon(x)$ is the closure of $B_\epsilon(x)$? Prove that the closure of any subset is always closed. \end{zadacha} \begin{zadacha} \label{_DISKRE_Zadacha_} Let $A$ be a subset of $M$ which has no accumulation points (such a subset is called {\bf discrete}). Prove that $M \backslash A$ is open. \end{zadacha} \begin{opredelenie} Let $M$ be a %compact (must've been a typo - it's too % early for compactness here metric space and $\epsilon > 0$ be a number. Consider $R\subseteq M$ such that $M$ can be covered by a union of all $\epsilon$-balls with center in $R$. Then $R$ is called an {\bf $\epsilon$-net}. \end{opredelenie} \begin{zadacha} Let any sequence in $M$ have an accumulation point. Prove that for any $\epsilon >0$ in $M$ there exists a finite $\epsilon$-net. \end{zadacha} \begin{ukazanie} Suppose that there is no such net, then for any finite set $R$ there exists a point $x$, whose distance to $R$ is more than $\epsilon$. Add $x$ to $R$, and, using this operation as induction step, obtain an infinite discrete subset of $M$. \end{ukazanie} \begin{opredelenie} Let $X\subset M$ and $U_i\subset M$ be a collection of open sets. If $X \subset \cup U_i$ then it is said that $U_i$ is a {\bf cover of $X$}. A collection of sets obtained from $\{U_i\}$ by throwing out some open sets in such a way that it remains a cover, is called a {\bf subcover}. \end{opredelenie} \begin{zadacha} \label{_shar_v_pokry_Zadacha_} Let $M$ be a metric space, $S$ be an open cover of $M$. Let every subsequence of elements of $M$ have an accumulation point. Prove that there exists such an $\epsilon>0$, that any ball of radius $<\epsilon$ is contained in one of the sets of the cover $S$. \end{zadacha} \begin{ukazanie} Suppose that for any $\epsilon$ there exists a point $x_\epsilon$ such that a corresponding $\epsilon$-ball is not contained entirely in any of the sets of the cover. Consider a sequence $\{\epsilon_i\}$ which converges to zero and let $x$ be an accumulation point of $\{x_{\epsilon_i}\}$. Prove that $x$ is not contained in any of the sets of $S$. \end{ukazanie} \begin{zadacha}[!]\label{comp.defn} (Bolzano-Weierstrass lemma) Let $X\subset M$ be a subset of a metric space. Prove that the following conditions are equivalent \begin{enumerate} \item Every sequence of points from $X$ has an accumulation point in $X$. \item Every open cover of $X$ has a finite subcover. \end{enumerate} \end{zadacha} \begin{ukazanie} Use problem~\ref{_DISKRE_Zadacha_} to deduce (a) from (b). In order to deduce (b) from (a), take an arbitrary cover $S$, a number $\epsilon$ from the problem~\ref{_shar_v_pokry_Zadacha_} and a finite $\epsilon$-net. Every ball of the $\epsilon$-net is contained in some of the elements $U_i\in S$. Prove that $\{U_i\}$ is a finite subcover. \end{ukazanie} \begin{opredelenie} Let $M$, $M'$ be metric spaces, and $f:\; M \to M'$ be a function. Then $f$ is called {\bf continuous}, if $f$ maps any sequence that converges to $x$ to a sequence that converges to $f(x)$, for all $x\in M$. \end{opredelenie} \begin{zadacha}[!] Let $X$ be any subset of $M$. Prove that a function $f:\; M \to \R$, $x \overset f \mapsto d(\{x\}, X)$ is continuous, where $d(\{x\}, X)$ (distance between $x$ and $X$) is defined as $d(\{x\}, X):=\inf_{x'\in X}d(x, x')$. \end{zadacha} \begin{opredelenie} Let $M$ be a metric space, $X\subset M$. It is said that $X$ is a {\bf compact set}, if any of the statements of the problem~\ref{comp.defn} holds. Note that these conditions do not depend on inclusion $X\hookrightarrow M$, but only on the metric on $X$. \end{opredelenie} \begin{zadacha}[!] Consider the completion of $\Z$ with respect to the norm $\nu_p$ defined above (it is called ``a ring of integer $p$-adic numbers'' and is denoted $\Z_p$). Prove that it is compact. \end{zadacha} \begin{ukazanie} Prove that any $p$-adic number can be represented in the from $\sum a_i p^i$, where $a_i$ are integers between 0 and $p-1$. \end{ukazanie} \begin{zadacha} Prove that a compact subset of $M$ is always closed. \end{zadacha} \begin{ukazanie} Prove that it contains all its accumulation points. \end{ukazanie} \begin{zadacha} Prove that a closed subspace of a compact set is always compact. \end{zadacha} \begin{zadacha} Prove that a union of a compact sets is compact. \end{zadacha} \begin{zadacha}[!] Let $f:\; X \to \R$ be a continuous function defined on a compact set. Prove that $f$ achieves maximum on $X$. \end{zadacha} \begin{opredelenie} Let $X$, $Y$ be two subsets of a metric space. Denote the number $\inf_{x\in X, y \in Y}(d(x,y))$ by $d(X, Y)$. \end{opredelenie} \begin{zadacha}[!] Let $X$, $Y$ be two compact subsets of a metric space. Prove that there exist points $x, y$ in $X, Y$ such that $d(x,y) = d(X,Y)$. \end{zadacha} \begin{opredelenie} A subset $Z\subset M$ is called bounded if it is contained in a ball $B_r(x)$ for some $r\in\R, x\in M$. \end{opredelenie} \begin{zadacha} Let $Z\subset M$ be compact. Prove that it is bounded. \end{zadacha} \begin{opredelenie} Let $M$ be a metric space and $X\subset M$. The union of all open $\epsilon$-balls with centers in all points of $X$ is called the {\bf $\epsilon$-neighbourhood} of $X$. \end{opredelenie} \begin{opredelenie} Let $M$ be a metric space and let $X$ and $Y$ be its bounded subsets. The {\bf Hausdorff distance} $d_{H}(X,Y)$ is the infimum of all $\epsilon$ such that $Y$ is contained in an $\epsilon$-neighborhood of $X$ and $X$ is contained in an $\epsilon$-neighborhood $Y$. \end{opredelenie} \begin{zadacha}[!] Prove that the Hausdorff distance defines a metric on the set $\cal M$ of all closed bounded subsets of $M$. \end{zadacha} \begin{zadacha} Let $X$, $Y$ be bounded subsets of $M$ and $x\in X$. Prove that it is always the case that $d_{H}(X, Y) \geq d(x, Y)$. \end{zadacha} \begin{zadacha}[!] Let $M$ be a complete metric space. Prove that $\cal M$ is also complete. \end{zadacha} \begin{ukazanie} Consider a Cauchy sequence $\{ X_i\}$ of subsets of $M$. Let $\mathfrak S$ be the set of Cauchy sequences $\{x_i\}$ with $x_i \in X_i$. Let $X$ be the set of accumulation points of sequences from $\mathfrak S$. Prove that $\{X_i\}$ converges to $X$. \end{ukazanie} \begin{zadacha}[*] Let $\{ X_i\}$ be a Cauchy sequence of compact subsets of $M$ and $X$ be its limit. Prove that $X$ is compact. \end{zadacha} \begin{ukazanie} One can identify $\{X_i\}$ with its subsequence such that \begin{equation}\label{eq:d_H} d_H(X_i, X_j)< 2^{-\min(i,j)}. \end{equation} Consider a sequence $\{x_i\}$ of points from $X$. For every $X_j$ find a sequence $\{x_i(j)\in X_j\}$ such that $d(x_i(j), x_i)= d(x_i, X_j)$. Since $X_j$ is compact, this sequence has an accumulation point. Choose an accumulation point $x(0)$ in $\{x_i(0)\}$ and replace $\{x_i\}$ with its subsequence such that $\{x_i(0)\}$ converges to $x(0)$. Then replace $\{x_i\}, i>0$ with a subsequence such that $\{x_i(1)\}$ converges to $x(1)$. We replace $\{x_i\}, i>k$ with a subsequence on $k$-the step in such a way that $\{x_i(k)\}$ converges to $x(k)$. Prove that we will finally obtain a sequence $\{x_i\}$ such that $\{x_i(k)\}$ converges to $x(k)$ for all $k$. Prove that this operation can be carried out in such a way that $d(x_i(k), x(k)) < 2^{-i}$. Use \eqref{eq:d_H} to prove that $d(x_i(k), x_i)< 2^{-\min(k,j)+2}$. Deduce that $\{x_i\}$ is a Cauchy sequence. \end{ukazanie} \begin{zadacha}[!] Let $M$ be compact and $X\subset M$. Prove that for any $\epsilon >0$ there is a finite set $R\subset M$ such that $d_H(R, X)<\epsilon$. (This statement can be rephrased as follows: ``$X$ allows approximation by finite sets with any prescribed accuracy'') \end{zadacha} \begin{ukazanie} Find a finite $\epsilon$-net in $X$. \end{ukazanie} \begin{zadacha}[*] Let $M$ be compact. Prove that $\cal M$ is also compact. \end{zadacha} \begin{ukazanie} Use the previous problem. \end{ukazanie} \begin{opredelenie} Let $M$ be a metric space. It is said that $M$ is {\bf locally compact}, if for any point $x\in M$ there exists a number $\epsilon>0$, such that the closed ball $\overline{B}_\epsilon(x)$ is compact. \end{opredelenie} \begin{zadacha} Let $M$ be a locally compact metric space and $\overline{B}_\epsilon(x)$ be a closed compact ball. Prove that $\overline{B}_\epsilon(x)$ is contained in an open set $Z$ with compact closure. \end{zadacha} \begin{ukazanie} Cover $\overline B_\epsilon(x)$ with balls such that their closures are compact, and find a finite subcover. \end{ukazanie} \begin{zadacha}[!] Prove in the previous problem setting that for some $\epsilon'> 0$ the ball $\overline{B}_{\epsilon+\epsilon'}(x)$ is also compact. \end{zadacha} \begin{ukazanie} Take $Z$ as in the previous problem. Take $\epsilon'$ to be $d (M \backslash Z, \overline B_\epsilon(x))$. \end{ukazanie} \begin{opredelenie} Let $(M, d)$ be a metric space. It is said that $M$ {\bf satisfies Hopf-Rinow condition} if for any two points $x, y\in M$ and for any two numbers $r_x, r_y >0$ such that $r_x+r_y < d(x,y)$ $$ d(B_{r_x}(x), B_{r_y}(y)) = d(x,y) - r_x-r_y. $$ \end{opredelenie} \begin{zadacha}[**] If you know the definition of a Riemannian (or Finsler) manifold, prove that the Hopf-Rinow condition holds for the natural metric on such a manifold. Justify all the facts that you use in the proof. \end{zadacha} \begin{zadacha}[*] Let $M$ be a complete locally compact metric space which satisfies Hopf-Rinow condition, $x\in M$ be a point and $\epsilon>0$ be a number such that $\overline B_{\epsilon'}(x)$ is compact for all $\epsilon'<\epsilon$. Prove that the ball $\overline B_{\epsilon}(x)$ is compact. \end{zadacha} \begin{ukazanie} Let $\{\epsilon_i\}$, with $\epsilon_i<\epsilon$, be a sequence that converges to $\epsilon$. Use the Hopf-Rinow condition to prove that $\{\overline B_{\epsilon_i}(x)\}$ is a Cauchy sequence with respect to Hausdorff metric, $\overline B_{\epsilon}(x)$. Use the fact that the limit of such a sequence is compact (you have already proved it before). \end{ukazanie} \begin{zadacha}[*] (Hopf-Rinow theorem, I) Let $M$ be a complete locally compact metric space which satisfies Hopf-Rinow condition. Prove that every closed ball $\overline B_{\epsilon}(x)$ in $M$ is compact. \end{zadacha} \begin{zadacha} Let $M$ be a metric space such that every closed ball $\overline B_{\epsilon}(x)$ in $M$ is compact. Prove that $M$ is complete. \end{zadacha} \begin{zadacha}[*] Let $M$ be a locally compact complete metric space which satisfies Hopf-Rinow condition, $x, y\in M$. Prove that there is a point $z\in M$ such that $d(x,z) = d (y, z)= \frac 1 2 d(x,y)$. \end{zadacha} \begin{zadacha}[*] Let $S$ be a set of all rational numbers of the form $\frac{n}{2^k}$, $n\in \Z$ which belong to the interval $[0,1]$. Prove in the previous problem setting that there exists a mapping $S\overset \xi\to M$ such that $d(\xi(a), \xi(b)) = |a-b| d(x,y)$ and $\xi(0)=x$ and $\xi(1)=y$. \end{zadacha} \begin{zadacha}[*] (Hopf-Rinow theorem, II) Let $M$ be a locally compact complete metric space which satisfies Hopf-Rinow condition, $x, y\in M$. Prove that the mapping $\xi$ can be naturally extended to the completion of $S$ with respect to the standard metric, so that the resulting mapping $[0,1]\overset {\overline\xi}\to M$ satisfies $\overline\xi(0)=x$, $\overline\xi(1)=y$ and $d((\overline\xi(a), \overline\xi(b)) = |a-b| d(x,y)$ for any two reals $a,b\in [0,1]$. \end{zadacha} \begin{zamechanie} Such a mapping $\overline\xi$ is called {\bf geodesic}. The Hopf-Rinow theorem can be restated as follows: for any two points in a complete metric locally compact space which satisfies Hopf-Rinow condition there is a geodesic that connects them. \end{zamechanie} \begin{opredelenie} Such a space is called {\bf geodesically connected}. \end{opredelenie} \begin{zadacha}[*] Give an example of a metric space, which is not locally compact but geodesically connected. \end{zadacha} \begin{zadacha} Let $V= \R^n$ be the metric space with the standard (Euclidean) metric. Prove that geodesics in $V$ are intervals (sets of the form $a x + (1-a) y$, where $a$ belongs to $[0, 1] \subset \R$, and $x, y \in V$). \end{zadacha} \begin{zadacha} Let $V$ be a finite dimensional vector space with a norm that defines a metric $d$ and $d_0$ be the Euclidean metric on $V$. Prove that the identity mapping $(V, d) \to (V, d_0)$ is continuous iff a unit ball in $(V, d)$ contains a ball from $(V, d_0)$. Prove that the inverse mapping is continuous provided that a unit ball in $(V, d)$ is contained in a ball from $(V, d_0)$. \end{zadacha} \begin{zadacha} In the previous problem settings, consider a function $D(x): = d(0,x)$ on a unit sphere $S^{n-1}\subset V $ $$ S^{n-1} = \{x\in V \mid d_0(0,x)=1\} $$ Let $D$ be a continuous function on $S^{n-1}$. Prove that the mapping $(V, d) \to (V, d_0)$ is continuous and the inverse mapping is continuous. \end{zadacha} \begin{ukazanie} Use the fact that a continuous function on a compact set achieves its minimum and maximum values. \end{ukazanie} \begin{zadacha}[**] Prove that $D$ is a continuous function. \end{zadacha} \begin{zadacha} Let $V$ be a finite dimensional vector space with a norm that defines the metric $d$. Suppose that the identity mapping $(V, d) \to (V, d_0)$ is continuous and the inverse mapping is also continuous. Prove that $(V, d)$ is complete and locally compact. \end{zadacha} \begin{zadacha}[*] Let $d$ be the metric on $\R^n$ associated with the norm $(x_1, x_2, ... ) \mapsto \max |x_i|$. Prove that it satisfies the Hopf-Rinow condition. Prove that $\R^n$ with such a metric is geodesically connected. Describe how the geodesics look like. \end{zadacha} \begin{zadacha}[*] Is it true that the metric $d$ defined by a norm always satisfies the Hopf-Rinow condition? \end{zadacha} \begin{opredelenie} Let $X$ be a metric space and $0