\documentclass[11pt]{article} \usepackage{geometry} % See geometry.pdf to learn the layout options. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry %\usepackage[parfill]{parskip} % Activate to begin paragraphs with an % empty line rather than an indent \usepackage{amsthm,graphicx} \usepackage{amssymb,amsmath} \usepackage{epstopdf} \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \def\R{{\mathbb R}}\textheight230truemm\textwidth160truemm \def\N{{\mathbb N}}\textheight230truemm\textwidth160truemm \def\T{{\mathcal T}} \def\cl{{\mathrm cl}} \renewcommand{\labelenumi}{{\bf \arabic{section}.\arabic{enumi}}} \renewcommand{\labelenumii}{(\alph{enumii})} \title{Topology problem set -- Integration workshop 2011} %\author{} %\date{} % Activate to display a given date or no date \begin{document} \maketitle %\section{} %\subsection{} \section{Topological spaces and Continuous functions} \begin{enumerate} \item If $\mathcal{T}_1$ and $\mathcal{T}_2$ are two topologies on $X$, show that $(X, \mathcal{T}_1 \cap \mathcal{T}_2)$ is also a topological space. Give an example where $\mathcal{T}_1 \cup \mathcal{T}_2$ is not a topology on $X$. \item {\bf The structure of open sets in $\mathbb{R}$.} In what follows, we are considering the standard (metric) topology on $\mathbb{R}$. \begin{enumerate} \item Let $S$ be a nonempty open subset of $\R$. For each $x \in S$, let $A_x = \{a \in \R : (a,x] \subseteq S\}$ and $B_x = \{b \in \R : [x,b) \subseteq S\}$. Show that, $A_x$ and $B_x$ are both non-empty. \item Where $x \in S$ as above, if $A_x$ is bounded below, let $a_x = \inf(A_x)$. Otherwise, let $a_x = - \infty$, and define $b_x$ is a corresponding manner. Show that $x \in I_x = (a_x,b_x) \subseteq S$. \item Show that $S = \cup_x I_x$. \item Show that the intervals $I_x$ give a partition of $S$, {\em i.e.}, for $x, y \in S$, either $I_x = I_y$ or $I_x \cap I_y = \emptyset$. \item Show that the set of distinct intervals $\{I_x : x \in S\}$ is countable. \item Prove that every open set in $\R$ is a countable disjoint union of open intervals. \end{enumerate} \item $(X,\T)$ is a topological spaces. \begin{enumerate} \item If $W\subseteq X$ is a subset, show that the \emph{relative} or \emph{subspace} topology of $W$ defined by $O\subseteq W$ is open only if $O=W\bigcap U$ for some $U\in\T$ is indeed a topology. \item Let $S_2 \subset S_1 \subset X$. Equip $S_1$ with the subspace topology. There are two ways to define a topology on $S_2$. We can give it the subspace topology it gets by thinking of it as a subset of $X$ or we can give it the subspace topology it gets by thinking of it as a subset of $S_1$. Show that there two topologies on $S_2$ are the same. \item Let $S_2 \subset S_1 \subset X$. We give $S_1$ the subspace topology. \begin{enumerate} \item Show that if $S_2$ is open in $S_1$ then it need not be open in $X$. Show that it is if $S_1$ is open in $X$. \item Show that if $S_2$ is compact in $S_1$, then $S_2$ is compact in $X$. \end{enumerate} \item If $(X,d)$ is a metric space, and $S\subseteq X$, then the restriction of $d$ to $S$ makes $S$ a metric space. We can define two topologies on $S$, the subspace topology on $S$ from the metric topology on $X$, and the metric topology on $S$ induced by the restriction of $d$ onto $S$. Show that the two topologies are the same. \end{enumerate} \item {\bf Product topology} Let $X$ be a collection of topological spaces indexed by a set $\mathcal{I}$. We define a topology on the Cartesian product $\prod_ \mathcal{\alpha \in I} X$ as follows: a basis is given by sets of the form $\prod_{\alpha \in \mathcal{I}} U_\alpha$, where $U_\alpha \subseteq X_{\alpha}$ is open and for all but finitely many $\alpha$ , $U_\alpha = X$ . \begin{enumerate} \item Prove that the above construction does indeed yield a basis. \item Prove that this is the weakest topology (with fewest open sets) such that the projections $\pi_\alpha: \prod_{\alpha \in \mathcal{I}} X_{\alpha} \rightarrow X_{\alpha}$ are continuous. \end{enumerate} \item {\bf Comparing topologies} $(X,\mathcal{T}_1)$ and $(X,\mathcal{T}_2)$ are topological spaces. The topology $\mathcal{T}_1$ is said to be {\em finer} than $\mathcal{T}_2$ if $\mathcal{T}_2 \subseteq \mathcal{T}_1$. \begin{enumerate} \item Show that $\mathcal{T}_1$ is finer than $\mathcal{T}_2$ if and only if for every $x \in X$ and every $U \in \mathcal{T}_2$ with $x \in U$, there is a $V \in \mathcal{T}_1$ with $x \in V$ such that $V \subseteq U$. \item Show that $\mathcal{T}_1$ is {\em finer} than $\mathcal{T}_2$ if and only if the identity map $\mathrm{Id}: (X,\mathcal{T}_1) \rightarrow (X,\mathcal{T}_2)$ is continuous. \item Show that the $l^1$ and $l^2$ metrics on $\mathbb{R}^2$ generate the same topology. \end{enumerate} \item If $f:(X,\mathcal{T}) \rightarrow (Y,\mathcal{S})$ and $g:(Y,\mathcal{S}) \rightarrow (Z,\mathcal{V})$ are continuous, show that the composition $g \circ f :(X,\mathcal{T}) \rightarrow (Z,\mathcal{V})$ is also continuous. \item A set in a topological space is {\em closed} if it's complement is open. If $f:X \rightarrow \mathbb{R}$ is a continuous function, show that $f^{-1}([0,1])$ is closed in $X$. %\item Problem 3.3.9, Pg. I-134 of Prof. Flaschka's notes. %\item Problems 3.3.10 and 3.3.11, Pg. I-134, Prof. Flaschka's notes. \item {\bf Local base for a topological space} \begin{enumerate} \item $\mathcal{N}(x)$ is a local base for a topological space $(X,T)$. If $A \in \mathcal{N}(x)$, show that there exists a $U \in T$ such that $x \in U \subseteq A$. \item $\mathcal{N}_1(x)$ and $\mathcal{N}_2(x)$ are local bases for a space $X$. Show that the topology $T_1$ generated by $\mathcal{N}_1(x)$ is finer that the topology $T_2$ generated by $\mathcal{N}_2(x)$ if and only if for all $B \in \mathcal{N}_2(x)$, there is a set $A \in \mathcal{N}_1(x)$ such that $x \in A \subseteq B$. %\item Let $X = l^{\infty}(\R,\N)$. For each $x \in X, m \in \N, \epsilon > 0$, let $U(x; m, \epsilon) = \{y \in X \, |\, \max_{1 \leq i \leq m} i^2 \, |x_i - y_i| < \epsilon\}$. If $\mathcal{N}(x) = \{U(x;m,\epsilon), m \in \N, \epsilon > 0\}$. Show that $\mathcal{N}(x)$ is a local base, and the $l^\infty$ metric topology on $X$ is strictly finer than the topology $T$ generated by this local base. %\item The topology $T$ is as defined in the previous item. Show that $x^{(n)} \in X$ converges to $z$ in $(X,T)$ if and only if $x_i^{(n)} \rightarrow z_i$ for all $i$. %\item Same setup as above. Let $V(x;\epsilon) = \{y \in X \, |\, \, |x_i - y_i| < \epsilon, \,\, \forall i \in \N\}$. If $\mathcal{M}(x) = \{V(x;\epsilon), \epsilon > 0\}$, is $\mathcal{M}$ a local base? \end{enumerate} % %\item {\bf Local base, base and subbase} %\begin{enumerate} %\item Exercise 3.4.7, Pg. I-138 in Prof. Flaschka's notes. (This is % the statement that we can define topologies through bases as % described in class) %\item Prove Proposition 3.4.13, Pg. I-139. (This is part of problem 3.4.12) %%\item Recall that a topological space is {\em second countable} if it has a countable base. Show that $\mathbb{R}$ with the usual topology is second countable. %%\item If $(X,\mathcal{T})$ is a second countable topological space, %% show that the topology is first countable. %%\item Problem 3.5.39, Pg. I-150 of Prof. Flaschka's notes. %%\item Show that $l^\infty$ with the metric topology is first countable, but not second %%countable. %\end{enumerate} \item {\bf First Countable local base} $(X,\mathcal{T})$ is first countable, if it has a local base $\mathcal{N}(x)$ such that at every point $x \in X$, the collection of neighborhoods $\mathcal{N}(x)$ is countable. \begin{enumerate} \item Show that the metric topology on a metric space $(X,d)$ is first countable. \item Is $(X,\mathcal{T})$ is first countable, show that every point $x \in X$ has a countable collection of open neighborhoods $U_n \ni x$ such that $U_n \subseteq U_{n+1}$, and $x$ is an interior point for an open set $O$ if and only if there is an index $n$ such that $x \in U_n \subseteq O$. \item With the same defintitions as the previous part, show that if you construct a sequence by picking arbitrary points $y_n \in U_n$, it follows that the sequence $\{y_n\}$ converges. \item If $(X,\mathcal{T})$ is first countable, and $(Y,\mathcal{S})$ is any topological space, show that $f:X \rightarrow Y$ is continuous if and only if it is sequentially continuous. \end{enumerate} \item {\bf Second countable spaces} Recall that a topological space is {\em second countable} if it has a countable base. \begin{enumerate} \item Show that $\mathbb{R}$ with the usual topology is second countable. \item Give an example of a space which is first countable but not second countable. \item Show that every second countable space is also first countable. \item A subspace $A$ of $X$ is called dense if the closure of $A$ is $X$. A topological space $X$ is called {\em separable}, if there exists a countable dense subset. Show that if $X$ is second countable, then $X$ is separable. \end{enumerate} \item A function $f: \R \rightarrow \R$ is {\em lower semi continuous} if for all $x \in \R, \epsilon > 0$, there exists a $\delta > 0$ such that $|y - x| < \delta$ implies that $f(y) > f(x) - \epsilon$. \begin{enumerate} \item Show that the collection $\mathcal{B} = \{ (\alpha, \infty) \, | \, \alpha \in \R \}$ is a base. Let $T'$ denote the topology generated by $\mathcal{B}$. Show that $T' \subset T_{metric}$ and the containment is strict. (Hint: One idea is to show that $T'$ is not Hausdorff.) \item Show that the collection $\mathcal{B}$ along with the empty set and all of $\R$ is the topology generated by the base $\mathcal{B}$, {\em i.e.} $T' = \mathcal{B} \cup \{\emptyset, \R\}$. \item Show that $T'$ is second countable. \item Show that a function $f : (\R,T_{metric}) \rightarrow (\R,T')$ is continuous, if and only if it is lower semi-continuous by the earlier definition. \item Show that a function $f : (\R,T') \rightarrow (\R,T_{metric})$ is continuous, if and only if it is a constant function. \end{enumerate} \item {\bf Accumulation points} $A \subseteq \R$, and $A'$ denotes the set of all the accumulation points of $A$. \begin{enumerate} \item If $y \in A'$ and $U \subseteq \R$ is an open set containing $y$, show that there are infinitely many distinct points in $A \cap U$. \item Show that $$ A' = \bigcap_{x \in A} \cl(A\backslash\{x\}). $$ \item Using this, or otherwise, show that $A'$ is a closed set. \item Show that $\cl(A) = A \cup A'$. \end{enumerate} %%\begin{enumerate} %%\item %$(X,T)$ is a topological space and the topology $T$ is Hausdorff. $A \subseteq X$. Let $A'$ denote the set of accumulation points of $A$. Show that %$$ %A' = \bigcap_{x \in A} \overline{A \backslash\{x\}}. %$$ %\end{enumerate} %\item {\bf Boundary points} %%\begin{enumerate} %\item Problem 3.5.30, Pg. I-150 of Prof. Flaschka's notes. %\item Show that the above definition of boundary point agrees with the definition in class, {\em viz.}, $x \in X$ is a boundary point of $A$ iff for every neighborhood $U$ of $x$, $U \cap A \neq \emptyset$ and $U \cap A^c \neq \emptyset$. %\end{enumerate} \item{\bf A topology on $\mathbb{N}$} Let $X = \mathbb{N} \cup \{e\}$. Define a collection $\mathcal{T}$ by $A \subseteq X$ is in $\mathcal{T}$ if and only if $A$ does not contain $e$ (this includes the empty set) or $e \in A$ and $A^c$ is finite (this includes $X$). \begin{enumerate} \item Show that $\mathcal{T}$ is a topology on $X$. \item Show that $\mathcal{T}$ is second countable. \item Show that $\mathbb{N}$ is dense in $(X,\mathcal{T})$. \item Is $(X,\mathcal{T})$ compact? \item A function $f:\mathbb{N} \rightarrow \R$ is the same thing as a sequence $x_n$. We will say that $g:X \rightarrow \R$ is a continuous extension of $f$ if $g(n) = f(n) \, \forall n \in \N$. Show that $f$ has a continuous extension iff $x_n = f(n)$ is a convergent sequence. Further, the continuous extension is given by $g(e) = \lim_{n \rightarrow \infty} f(n)$. \item Every element $a = (l,a_1,a_2,a_3,\ldots) \in\R \times \R^\N $ defines a function $f_a:X \rightarrow \R$ by $f_a(n) = a_n, f_a(e) = l$. Let $Y \subset \R \times \R^\N$ denote the set of all the convergent sequences with their associated limits, {\em i.e.} $(l,a_1,a_2,a_3,\ldots) \in Y \implies a_n \rightarrow l$. Find the weakest topology on $X$ such that for all $a \in Y$, $f_a:X \rightarrow \R$ is continuous. \item Can you find a metric on $X$ such that the metric topology is identical to the topology $\mathcal{T}$ above? Any topology with this property is said to be {\em metrizable}. \end{enumerate} \item A topological space $X$ is called a T1-space (or a Tychonoff space) if for any two different points $x, y \in X$ there exists an open set $U$ which contains $x$ but does not contain $y$. Prove that a space $X$ is a T1-space if and only if any subset consisting of a single point is closed. Also find an example of a topological space which is a T1-space, but not Hausdorff (a T2-space). \item Define the profinite topology on $\mathbb{Z}$ in which the open sets are the empty set and unions of arithmetic progressions. \begin{enumerate} \item Show that an arithmetic progression is also a closed set in this topology. \item Show that if there were only finitely many primes, then the set $\{-1, 1\}$ would be open. \item Then show that this set is not open and conclude that there are infinitely many primes. \item Let $T^\infty$ be the product of countably infinitely many copies of the unit circle with the product topology. Define the map $\phi: \mathbb{Z} \rightarrow T^\infty$ as follows: $$ \phi(n) = \left(\exp(2 \pi i n/2), \exp(2 \pi i n/3), \exp(2 \pi i n/4), \exp(2 \pi i n/5), ...\right). $$ Show that this map is injective and the induced topology on $\mathbb{Z}$ coincides with the profinite topology. \end{enumerate} %\item A topological space X is said to satisfy the first axiom of countability if for each point %x 2 X there is a countable basis for the complete system of neighbourhoods at x (i.e. %each neighbourhood contains a neighbourhood from the countable basis). Show that if X %has a countable basis of its topology (i.e. is second countable), then it is first countable %as well. %\item Show that the product of copies Q of the unit interval indexed by the unit interval, % 2[0,1][0, 1] is not first countable. \item {\bf Zariski topology} Consider the topology on $\R^n$ in which the open sets are the empty set and the complements of the common zero levels sets of finitely many polynomials. Show that this is indeed a topology on $R^n$. This is called the Zariski topology. Show also that the Zariski topology is not Hausdorff. \item Let the group $\R$ act on $\R^2$ by $$t.(x, y) = (x, y + tx) . $$ Prove that the quotient space with the quotient topology is not Hausdorff, but is the union of two disjoint Hausdorff subspaces. Also show that the quotient space is a T1-space. \item \begin{enumerate} \item Show that $\R$ and $(0,1)$ are homeomorphic. \item Let $f : (0,1) \rightarrow \R$ be your homeomorphism. Show there is a Cauchy sequence $x_n$ in $(0,1)$ such that $f(x_n)$ is not Cauchy in $\R$. \end{enumerate} \item Show that the line with two origins is not Hausdorff. \item In the lectures we stated a proposition that said that if $(X,d)$ and $(Y,d^\prime)$ are metric spaces and $f:X \rightarrow Y$, then the $\epsilon-\delta$ definition of continuity of $f$ and the open set defintion are equivalent. Prove this proposition. \item Find a topological space $X$ and a sequence $x_n$ in $X$ which converges but has more than one limit. What additional property on $X$ implies that limits of sequences are unique? \end{enumerate} \section{Compactness} \begin{enumerate} \item Prove or disprove the following: \begin{enumerate} \item $A$ is finite and $U$ is a open subset of $\R$. If $A \subseteq U$, there exists an $\epsilon > 0$ such that for all $x \in A$, $N(x,\epsilon) \subseteq U$. \item $P$ is countable and $U$ is a open subset of $\R$. If $P \subseteq U$, there exists an $\epsilon > 0$ such that for all $x \in P$, $N(x,\epsilon) \subseteq U$. \item $F$ is closed and $U$ is a open subset of $\R$. If $F \subseteq U$, there exists an $\epsilon > 0$ such that for all $x \in F$, $N(x,\epsilon) \subseteq U$. \item $K$ is compact and $U$ is a open subset of $\R$. If $K \subseteq U$, there exists an $\epsilon > 0$ such that for all $x \in K$, $N(x,\epsilon) \subseteq U$. \end{enumerate} %\item Show that a map between two metric spaces $f : (X, d) \rightarrow (Y, d0)$ is continuous at a %point $p \in X$ if for any neighbourhood $M$ of $f(p)$ its pre-image $f^{-1}(M)$ is a neighbourhood %of $p$. \item If $X$ is compact and $f:X \rightarrow Y$ is continuous, show that $f(X)$ is compact. \item Prove that the unit sphere in $\R^n$ is compact. \item Consider the topology on $X$ in which the open sets are the empty set and the complements of finite subsets. Show that every subset of $X$ is compact, although not every subset of $X$ is closed, in general. \item Let $\R\mathbb{P}^n$ denote the quotient space of $\R^{n+1} \setminus \{0\}$, by the equivalence relation $x\sim y$ iff $\exists \lambda \neq 0$, s.t. $x = \lambda y$. Show that $\R\mathbb{P}^n$ is compact. \item Show that every compact subset of a Hausdorff space is closed. \item Show that if X is compact and Y is Hausdorff and $f : X \rightarrow Y$ is a continuous bijection, then f is a homeomorphism. \item A space is {\em locally compact} if every point has a compact neighbourhood. \begin{enumerate} \item If $X$ is a compact space, then show that $X$ is locally compact. \item Give an example of a space which is not locally compact. \item There is a canonical way to add one point to a locally compact Hausdorff space to get a compact space. Namely, if $X$ is locally compact Hausdorff, let $\bar{X} = X \cup \{\infty\}$. The open sets of $\bar{X}$ are the open sets of $X$ together with the sets $(X \setminus K) \cup \{\infty\}$, where $K$ is a compact subset of $X$. Prove that $\bar{X}$ , called the {\em one point compactification} of $X$, is a compact Hausdorff space. \end{enumerate} %\item In a metric space $(X, d)$, a sequence a1, a2, ... of points of %X is called a Cauchy sequence %if for each " > 0 there is a positive integer N such that % d(an, am) < " whenever n,m > N. %A metric space is called complete if every Cauchy sequence in X %converges to a point of X. Show that a compact metric space is complete. \item A metric space $X$ is said to be {\em totally bounded} if for any $\epsilon > 0$ it can be covered by finitely many $\epsilon$-balls. Also, a metric space is complete, if every Cauchy sequence in $X$ converges. Prove that $X$ is compact if and only if $X$ is complete and totally bounded. \item $l^2$ is the space of square summable sequences of real numbers, i.e., $(x_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty x_n^2 < \infty$. The norm is defined by \begin{equation} || (x_n)_{n=1}^\infty|| = \left[ \sum_{n=1}^\infty x_n^2 \right]^{1/2} \end{equation} \begin{enumerate} \item Prove the interior of every compact subset of $l^2$ is empty. \item Define \begin{equation} F= \{ (x_n)_{n=1}^\infty : \sum_{n=1}^\infty \, n \, x_n^2 \le 1\} \end{equation} Prove that $F$ is sequentially compact. \end{enumerate} \item Prove that if a topological space is compact then it is sequentially compact. \end{enumerate} \section{Connectedness} \begin{enumerate} \item Let $X$ be the union of the origin in $\R^2$ and the graph of $sin(1/x)$ on $(0,\infty)$. Show $X$ is connected but not path connected. \item Show that the Cantor set is totally disconnected. \item If $X$ is connected and $f:X \rightarrow Y$ is continuous, show that $f(X)$ is connected. \item Find all the different topologies, up to homeomorphism, on a 4-element set, which make it a connected topological space. \item Prove that the closure of a connected subspace is connected. \item Show that $\R$ and $\R^2$ are not homeomorphic. Hint: use the notion of a connected set. \item Prove that each connected component of a topological space $X$ is closed. \item Show that if $A$ is a both open and closed, non-empty, connected subset of a topological space $X$, then $A$ is a connected component. \item Show that if a topological space has finitely many connected components, then each of them is open and closed. \item A space $X$ is called locally path-connected, if for each $x \in X$ and every neighbourhood $U$ of $x$, there exists a path-connected neighbourhood $V$ of $x$ contained in $U$. Show that if $X$ is connected and locally path-connected, then it is path-connected. \item Show that if $K$ is the Cantor set, then the complement of $K \times K$ in the unit square $[0, 1] \times [0, 1]$ is path-connected. \end{enumerate} \end{document}