\documentclass[11pt,notitlepage]{report} \frenchspacing % aktivuje použití některých českých typografických pravidel \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} % nastavuje použité kódování, uživatelé Windows zamění utf8 za cp1250 \usepackage{amsthm,amsmath,amssymb,amscd,mathrsfs,enumerate,enumitem,multicol,color,url}%,showkeys,layout} \usepackage{verbatim} %pro vynechavani comment \usepackage[top=1cm, left=1cm, right=1cm]{geometry} % nastavení dané velikosti okrajů \pagestyle{empty} \theoremstyle{definition} \newtheorem*{defin}{Definice} \def\ce{\mathbb C} \def\en{\mathbb N} \def\zet{\mathbb Z} \def\qe{\mathbb Q} \def\er{\mathbb R} \def\HH{\mathbb H} \def\A{\mathcal A} \def\U{\mathcal U} \def\B{\mathcal B} \def\C{\mathcal C} \def\D{\mathcal D} \def\R{\er} \def\N{\en} \def\setsep{:\;} \def\nec{Nech\v t\ } \def\eqdef{\stackrel{\text{\tiny def}}{=}} \def\Mod{\operatorname{mod}} \def\lp{\displaystyle\lim_{n\to\infty}} \newcommand\lx[1][0]{\displaystyle\lim_{x\to #1}} \def \cotg {\operatorname{cotg}} \def \ind {\operatorname{ind}} \def \sgn {\operatorname{sgn}} \def \Ind {\operatorname{Ind}} \newcommand{\cl}{\overline } \def\Int{\operatorname{Int}} \def \suma {\sum_{n=1}^\infty} \def \sumad {\sum_{n=2}^\infty} \def \dx {\, \mathrm{d} x} \def \P {\mathcal{P}} \def\x{\mathbf{x}} \def\m{\mathbf{m}} \newcommand\dgx[1]{\, \mathrm{d} #1} \def\vysledky{\begin{center}----------V\'YSLEDKY----------\end{center}} \newtheorem*{thm}{V\v{e}ta} \newcounter{Priklad} \addtocounter{Priklad}{1} \newcommand{\pr}{\quad\alph{Priklad})\;\addtocounter{Priklad}{1}} \newcommand{\noindentpr}{\alph{Priklad})\;\addtocounter{Priklad}{1}} \newcommand{\resetpr}{\setcounter{Priklad}{1}} \title{Cvi\v{c}en\'i - Obecná topologie} % tyto dvě položky jsou zde v podstatě formálně, ve skutečnosti nejsou nikde \author{Marek C\'uth} \begin{document} % Cviceni rozdam vzdy kdyz zacneme probirat nejakou kapitolu. Cokoliv ze cviceni muze byt pouzito u zkousky jakozto "priklad navic". Zapocet je zadarmo (student jej dostane zaroven se zkouskou). Idea jak jednotliva cviceni vyresit se bude probirat vzdy az po dokonceni prislusne kapitoly, vyhradi se na to jedna 90ti minutovka (na celou serii prislusnou jedne kapitole) .. bude probihat jako moderovane setkani, kde studenti rikaji jak se snazili jednotlive ulohy resit a ja k tomu pripadne reknu jak jsem to resil ja. \centerline{\MakeUppercase{Exercises - Uniform Spaces}}\ \\ \textbf{1. Find two uniformities $\D_1$ and $\D_2$ on $\R$ which both generate the usual topology on $\R$, yet there exists no uniform homeomorphism between $(\er,\D_1)$ and $(\er,\D_2)$.} \medskip \noindent\textbf{2. Let $(X,\mathcal U)$ be a uniform space. Prove that if $(X,\D)$ is metrizable, then $(X,\tau_\D)$ is metrizable, and find an example showing that the converse implication does not hold.} \medskip \noindent\textbf{3. Uniformity given by a system of coverings: } Let $X$ be a set. A system $\mathscr U$ is called a \emph{covering uniformity} on the set $X$ if: \begin{itemize} \item[(a)] Every $\mathcal U \in\mathscr U$ is a covering of $X$. \item[(b)] If $\mathcal U\in\mathscr U$ and $\mathcal U$ is a refinement of some covering $\mathcal V$, then $\mathcal V\in\mathscr U$. \item[(c)] For $\mathcal U, \mathcal V\in \mathscr U$ it holds that $\mathcal U\wedge \mathcal V\in\mathscr U$. \item[(d)] For every $\mathcal U\in\mathscr U$ there exists $\mathcal V\in\mathscr U$ such that $\mathcal V$ strongly star-refines $\mathcal U$. \end{itemize} \noindent(If $X$ is a set and $\mathcal U$ is a covering of $X$, then a system $\mathcal V$ is called a \emph{refinement} of $\mathcal U$ if $\mathcal V$ is a covering of $X$ and for every $V\in\mathcal V$ there exists $U\in\mathcal U$ such that $V\subseteq U$. For a system $\mathcal S$ of subsets of $X$ and $A\subseteq X$ define $st_{\mathcal S}(A)=\bigcup\{S\in\mathcal S\colon S\cap A\neq\emptyset\}$. We say that a covering $\mathcal U$ \emph{strongly star-refines} $\mathcal V$ if $\{st_{\mathcal U}(U)\colon U\in\mathcal U\}$ refines $\mathcal V$. For two coverings $\mathcal U, \mathcal V$ we define their common refinement $\mathcal U\wedge\mathcal V$ as $\{U\cap V\colon U\in\mathcal U, V\in\mathcal V\}$.)\\[3pt] If $\mathscr U$ is a covering uniformity, set \[\mathcal D_{\mathscr U} =\{D\subseteq X\times X\colon\exists \mathcal U\in\mathscr U\colon \bigcup\{U\times U\colon U\in \mathcal U\} \subseteq D \},\] and if $\mathcal D$ is a uniformity, set \[\mathscr U_{\mathcal D}=\{\mathcal U \in\mathcal P(\mathcal P(X))\colon\exists D\in\mathcal D: \{D[x]: x\in X\}\text{ refines } \mathcal U\}.\] Prove that the mapping $\mathcal U\mapsto \D_{\mathcal U}$ is a bijection between covering uniformities and uniformities, where the inverse mapping is given by $\D\mapsto \mathcal U_{\D}$. \medskip \noindent\textbf{4. Topology on a subspace and product: }Prove that if $(X,\D)$ is a uniform space and $A\subset X$, then the topology on $A$ corresponding to $\D|_A$ coincides with the subspace topology. Furthermore, prove that if $(X_i,\D_i)$, $i\in I$ are uniform spaces, then the topology on $\Pi_I X_i$ corresponding to $\D_{\Pi_I X_i}$ is the product topology. \medskip \noindent\textbf{5. Existence of completion: } Let $(X,\D)$ be a $T_1$ uniform space where $\D$ is generated by a system of pseudometrics $R$. \begin{itemize} \item Denote $\widetilde{X}_0:=\{(x_i)_{i\in I}\colon (x_i) \text{ is a Cauchy net}\}$. Whenever $d\in R$, consider $\widetilde{d}_0:\widetilde{X}_0\times \widetilde{X}_0\to [0,\infty)$ defined by \[ \widetilde{d}_0\big((x_i),(y_j)\big):= \lim_{(i,j)\in I\times J} d(x_i,y_j),\qquad \text{for }(x_i)_{i\in I}\in \widetilde{X}_0\text{ and }(y_j)_{j\in J}\in \widetilde{X}_0 \] (on $I\times J$ we consider the ordering $(i,j)\leq (i',j')$ iff $i\leq i'$ and at the same time $j\leq j'$).\\ Prove that $\widetilde{d}_0$ is a well-defined pseudometric on $\widetilde{X}_0$. \item For $x\in \widetilde{X}_0$ consider $[x]:=\{y\in \widetilde{X}_0\colon \widetilde{d}_0(x,y)=0\text{ for every }d\in R\}$. Consider on the space $\widetilde{X}:=\{[x]\colon x\in \widetilde{X}_0\}$ the system of pseudometrics $\widetilde{R}:=\{\widetilde{d}\colon d\in R\}$, where for $d\in R$ the pseudometric $\widetilde{d}$ is given by $\widetilde{d}([x],[y]):=\widetilde{d}_0(x,y)$. Prove that $\widetilde{R}$ is a well-defined system of pseudometrics on $\widetilde{X}$ and that $(\widetilde{X},\widetilde{R})$ is a $T_1$ uniform space. \item Define a mapping $e:X\to \widetilde{X}$ by $e(x):=[x]$ (the point $x$ is identified with the net consisting of a single point). Prove that $e$ is a uniform embedding and that $e(X)$ is dense in $\widetilde{X}$. \item Prove that $\widetilde{X}$ is complete.\\ (\emph{Hint: it is useful to first realize that if $D$ is a dense subset of a uniform space $M$, then $M$ is complete if and only if every Cauchy net consisting of points in $D$ converges in $M$.}) \end{itemize} \newpage \centerline{\MakeUppercase{Exercises - Topological Groups}}\ \\ \textbf{1. The topological group $H(K)$ – basics: }Let $K$ be a $T_2$ compact space.\ \resetpr \noindentpr Prove that $H(K)$ is a topological group.\ (Here the symbol $H(K)$ denotes the set of all surjective homeomorphisms $f:K\to K$ with the group operation of composition and the \emph{compact-open} topology, i.e. the topology whose subbasis consists of sets $E[L;U]:={f\in H(K)\colon f(L)\subset U}$, where $L\subset K$ is compact and $U\subset K$ is open.)\ \noindentpr Prove that if $(K,d)$ is a metric compact space, then the compact-open topology on $H(K)$ is metrizable by the metric $\rho$ defined by $\rho(f,g):=\sup_{x\in K} d(f(x),g(x))$ for $f,g\in H(K)$. \medskip \noindent\textbf{2. Topology on a group generated by a system of bi-invariant pseudometrics.} Let $G$ be a group and let $R$ be a system of bi-invariant pseudometrics separating points. Then the \emph{topology generated by the system $R$} is the topology $\tau_R$, whose neighborhood basis at a point $x$ has a subbasis generated by ${B_\rho(x,\varepsilon)\colon \varepsilon>0,; \rho\in R}$.\ \resetpr \noindentpr Prove that $G$ with the topology $\tau_R$ is a $T_1$ topological group which is SIN.\ \noindentpr Prove that a $T_1$ topological group $(G,\cdot,\tau)$ is SIN if and only if there exists a system $R$ of bi-invariant pseudometrics satisfying $\tau = \tau_R$. \medskip \noindent\textbf{3. Topological groups that are / are not SIN: }\ \resetpr \noindentpr Consider the topological group $G = {(a,b)\colon a>0,b\in\R}$ with the group operation defined by $(a,b)\cdot(a',b'):=(aa',b+ab')$ and with the topology inherited from $\R^2$. Prove that $G$ is a metrizable topological group which is not metrizable by a bi-invariant metric.\ (\emph{Hint for proving that it is not metrizable by a bi-invariant metric: For a contradiction assume that $G$ is metrizable by a bi-invariant metric $\rho$. Show that then there exists $\varepsilon>0$ and $t\neq 0$ satisfying $(1,t)\in B_\rho(e,\varepsilon)\subset {(a,b)\colon |b|<1}$, deduce that then $(1,at)\in B_\rho(e,\varepsilon)$ for every $a>0$, which yields a contradiction.}) \smallskip \noindent\noindentpr Decide and prove whether the metrizable topological group $H([0,1])$ is SIN.\ (For the definition and basic properties of $H([0,1])$ see Example 1 above). \smallskip \noindent\noindentpr Consider the group $S_\infty$ (permutations of $\N$ with the operation of composition) with the topology of pointwise convergence (i.e. basic neighborhoods of $\pi\in S_\infty$ consist of sets $U[\pi;F]:={\sigma\colon \sigma|F = \pi|F}$, where $F\subset \N$ are finite subsets). Prove that then $S\infty$ is a metrizable topological group. Decide and prove whether $S\infty$ is SIN. \medskip \noindent\textbf{4. Closed subgroups and quotients of $\R$:}\ On $\R$ consider the operation $+$ and the standard topology generated by the absolute value.\ \resetpr \noindentpr Find all closed subgroups of $\R$.\ \smallskip \noindentpr Show that every quotient of $\R$ by a closed subgroup is, as a topological group, isomorphic to ${0}$, $\R$, or $\mathbb{T}$.\ ($\mathbb{T}$ denotes the circle ${e^{ix}\colon x\in \R}\subset \ce$ with the operation $e^{ix}e^{iy}=e^{i(x+y)}$ and the topology inherited from $\ce = \R^2$). \medskip \noindent\textbf{5. Find an example of a $T_1$ topological group which is not $T_4$.} % for example Z^R .. the product of topological groups is a topological group, but Z^R is homeomorphic to N^R and this is not a normal topological space, because it is separable (must be proved) and contains a closed discrete subset of cardinality continuum .. or one can take Z^{\omega_1} and use the proof from the script, see Example 2.38 % examples of topological spaces which are / are not topological groups \medskip %\noindent\textbf{6. Every $T_1$ topological group embeds as a topological group into $H(K)$ for some $T_2$ compact space $K$.} \resetpr %\noindentpr Let $V$ be a Banach space and consider $K:=(B_{V^*},w^*)$. Show that then the mapping $\operatorname{Iso}(V)\ni T\mapsto T^*|_{B_{V^*}}\in H(K)$ is an embedding of topological groups.\ %\noindentpr Using Teleman’s theorem deduce that every $T_1$ topological group embeds as a topological group into $H(K)$ for some $T_2$ compact space $K$. % it should not be too difficult .. see Megrelishvili’s notes for possible details .. \newpage \centerline{\MakeUppercase{Exercises - Paracompactness}}\ \\ \textbf{1. Preservation of paracompactness: } \begin{enumerate}[label=(\alph*)] \item Prove that an $F_\sigma$ subset of a paracompact space is paracompact. \item Find an example of a subset of a paracompact space which is not paracompact. \item Find an example of a paracompact space $X$ such that $X\times X$ is not paracompact. \item Prove that the product of a paracompact space and a compact $T_2$ space is paracompact. \item A topological space is called $\sigma$-compact if it is the union of countably many of its compact subspaces. Prove that the product of a paracompact and a $\sigma$-compact $T_{3\frac 12}$ space is paracompact.\ (Observe that as a consequence we obtain that for every Banach space $X$, the space $(X^*,w^*)$ is paracompact, since $(X^*,w^*)$ is $\sigma$-compact.) \item Prove that $[0,\omega_1)$ is collectionwise normal but not paracompact.\ (\emph{Hint: }Prove that in a countably compact space every locally finite cover is finite. Deduce that every paracompact space which is countably compact is already compact. However, $[0,\omega_1)$ is countably compact and not compact.) \end{enumerate} \medskip \noindent\textbf{2. Metric spaces embed into a countable product of hedgehogs: }For a set $I$ define the \emph{metric hedgehog with $I$ spines} as the metric space $(I\times [0,1])/{\sim}$, where $(i,x)\sim (j,y)$ iff $x=y=0$, equipped with the metric \[d((i,x),(j,y)):=\begin{cases}|x-y| & i=j,\\ x + y & i\neq j.\end{cases}\] Prove that every metric space embeds into a countable product of metric hedgehogs.\\ (\emph{Hint: try to be inspired by the proof of the Bing–Nagata–Smirnov theorem}) \iffalse \medskip \noindent\textbf{2. Characterization of paracompactness using partitions of unity: }Prove that a $T_{3\frac 12}$ space is paracompact if and only if for every open cover of $X$ there exists a locally finite partition of unity subordinate to this cover. \medskip \noindent\textbf{4. Separation by Lipschitz functions: }Let $M$ be a metric space satisfying $\operatorname{dens} M\leq 2^\omega$. Prove that there exist countably many Lipschitz functions separating the points of $M$. % Add also a hint according to our paper with Ben \fi \newpage \centerline{\MakeUppercase{Exercises - Connectedness}}\ \\ \textbf{1. Cardinality of a connected space: }\resetpr \pr Prove that a connected $T_{3 \frac 1/2}$ space with at least two points has cardinality at least continuum. \pr Prove that a connected $T_{3}$ space with at least two points is uncountable. \pr Find an example of an infinite connected $T_2$ space which is countable.\\ (\emph{Hint: first find a system of pairwise disjoint subsets of rational numbers $\qe_n$, $n\in \en\cup \{0\}$ such that each $\qe_n$ is dense in $\er$. Then consider $X =\{\infty\}\cup \bigcup_{n\in\en\cup\{0\}} \qe_n\times \{n\}$, where a neighborhood basis of $\infty$ consists of sets $\{\infty\}\cup \bigcup_{n\geq k} \qe_n\times \{n\}$ for $k\in \en$, a neighborhood basis of $(q,k)$ for $q\in \qe_k$ and $k$ odd is given by $X\cap \big((q-\varepsilon,q+\varepsilon)\times \{k-1,k,k+1\}\big)$ and a neighborhood basis of $(q,k)$ for $q\in \qe_k$ and $k$ even is given by $X\cap \big((q-\varepsilon,q+\varepsilon)\times \{k\}\big)$. Verify that then $X$ is countable, $T_2$, and whenever a clopen set $C\subset X$ contains $\infty$ then $C = X$.}) \medskip \noindent\textbf{2. Connected linearly ordered topological spaces: }\resetpr Prove that a linearly ordered topological space $(X,\leq)$ is connected if and only if every nonempty bounded above subset has a supremum and $(x,y)\neq \emptyset$ for every $x,y\in X$. \medskip \noindent\textbf{3. Connected components of $H([0,1])$: }Describe all connected components of the topological group $H([0,1])$.\\ (For the definition and basic properties of $H([0,1])$ see Exercise 1 on topological groups.) \medskip \noindent\textbf{4. Connected components of $GL(n,\er)$: }Describe all connected components of the topological group $GL(n,\er)$. \medskip \noindent\textbf{5. Inverse limits and examples of continua: }\emph{An inverse sequence} is a sequence of pairs $(X_n, f_n)_{n=1}^\infty$, where $X_n$ are topological spaces and $f_n:X_{n+1}\to X_n$ are continuous mappings. The \emph{inverse limit} of the inverse sequence $(X_n, f_n)_{n=1}^\infty$ is the following subspace $\lim_{\leftarrow} (X_n, f_n)$ of the product $\prod_\en X_n$: \[ \lim_{\leftarrow} (X_n, f_n):=\{(x_n)_{n\in\en}: f_n(x_{n+1}) = x_n \text{ for each }n\in\en\}. \] (Sometimes we also denote it by $\lim_{\leftarrow} X_n$ or $X_\infty$, if it is clear from the context what $X_n$ and $f_n$ are.)\\ \noindentpr Prove that if $X_n$ are Hausdorff compact spaces (resp. continua), then $X_\infty$ is a Hausdorff compact space (resp. a continuum).\\ \noindentpr Find an inverse system consisting of finite discrete spaces whose inverse limit is homeomorphic to the Cantor discontinuum.\\ \noindentpr An inverse sequence $(X_n, f_n){n=1}^\infty$ is called \emph{indecomposable} if for every $i\in \en$ and every two continua $A,B\subset X{i+1}$ whose union is $X_{i+1}$, it holds that $f_i(A) = X_i$ or $f_i(B) = X_i$. Prove that if $X_n$ are continua and $(X_n, f_n){n=1}^\infty$ is an indecomposable inverse sequence, then $X\infty$ is an indecomposable continuum.\\ \noindentpr Let $X_n=[0,1]$ and $f_n(x) = |1-2x|$ for every $n\in\en$. Prove that $X_\infty$ is then an indecomposable continuum.\\ (\emph{Note: this is the so-called Knaster continuum.})\\ %\noindentpr Let $X_n=\mathbb{T} = {z\in\ce\colon |z|=1}$ and $f_n(z) = z^2$ for every $n\in\en$. Prove that $X_\infty$ is then an indecomposable continuum on which it is possible to define a multiplication operation such that $X_\infty$ is also a commutative topological group.\\ %(\emph{Note: this is the so-called dyadic solenoid.}) \medskip \noindent\textbf{6. Strongly zero-dimensional metric spaces: }We say that a metric $\rho$ on $X$ is an \emph{ultrametric} if $\rho(x,y)\leq \max{\rho(x,z),\rho(z,y)}$ for every $x,y,z\in X$. In that case we say that $(X,\rho)$ is an \emph{ultrametric space}. We say that a topological space is \emph{ultrametrizable} if it is homeomorphic to an ultrametric space. Prove the following.\\ \resetpr \noindentpr Every ultrametric space is zero-dimensional. (\emph{Hint: open balls are closed sets.})\\ \noindentpr The space $2^\omega$ is ultrametrizable. Deduce from the theorem from the lecture that a separable metric space is zero-dimensional if and only if it is ultrametrizable.\\ \emph{The following exercises aim to show that an analogue of the previous statement also holds for nonseparable spaces. There exists an example (rather complicated)% see \emph{P. Roy: Nonequality of dimensions for metric spaces, 1968}) of a nonseparable metric space which is zero-dimensional and not strongly zero-dimensional. It turns out that the correct notion is that of strong zero-dimensionality.}\\ \noindentpr A metrizable space is strongly zero-dimensional if and only if it is ultrametrizable.\\ \medskip \noindent\textbf{7. An example of a normal subspace of a compact space which is zero-dimensional but not strongly zero-dimensional: }\\ \resetpr \noindentpr Prove that there exists a decomposition ${A_\alpha\colon \alpha<\omega_1}$ of the interval $[0,1]$ such that for every $\alpha<\omega_1$ the set $A_\alpha$ is dense in the interval $[0,1]$.\ \noindentpr For $\alpha<\omega_1$ denote $B_\alpha=\bigcup{A_\beta\colon \beta\leq \alpha}$, where the sets $A_\alpha$ are as above. Let $D\subseteq [0,1]\times \omega_1$ be defined by \[D=\{(r,\alpha)\colon r\in B_\alpha, \alpha<\omega_1\}.\] Prove that the space $D$ is zero-dimensional.\\ \noindentpr Let $D\subset [0,1]\times [0,\omega_1]$ be as above. Denote $D^*:=D\cup ([0,1]\times{\omega_1})$. Prove that $D^*$ is a normal space.\\ \noindentpr Let $D$ and $D^*$ be as above. Prove that whenever $A,B$ are disjoint closed subsets of $D$, then their closures in $D^*$ are also disjoint.\\ \noindentpr Let $D\subset [0,1]\times [0,\omega_1]$ be as above. Using all the information above prove that $D$ is a normal space which is zero-dimensional but not strongly zero-dimensional. \newpage \centerline{\MakeUppercase{Exercises - Dimension}}\ \\ Let $X$ be $T_4$. Recall that $A\subset X$ is a \emph{zero} (resp. \emph{cozero}) set if there exists a continuous function $f:X\to \er$ satisfying $A = f^{-1}(0)$ (resp. $A = f^{-1}(\er\setminus{0})$). We say that $M\subset X$ is \emph{$C^*$-embedded} if every continuous function $f:M\to [0,1]$ has a continuous extension to a continuous function $\widetilde{f}:X\to [0,1]$.\\ \medskip \noindent\textbf{1. Dimension of a subspace: }\resetpr \\ \noindentpr Using the previous exercises find an example of a compact space $X$ and its normal subspace $M$ such that $\Ind M > \Ind X$ and also $\dim M > \dim X$.\\ \noindentpr Let $X$ and $M$ be $T_4$ spaces and $M\subset X$ be $C^*$-embedded. Prove that then $\dim M\leq \dim X$.\\ \noindentpr Let $D$ and $D^*$ be as in the Exercises on Connectedness, Example 7. Prove that $D\subset D^*$ is $C^*$-embedded and using the previous result deduce that $\dim D \leq \dim D^*$.\\ \noindentpr Let $D$ and $D^*$ be as in the Exercises on Connectedness, Example 7. Prove that $\operatorname{ind} D = 0 < 1 = \dim D \leq \Ind D$.\\ (Automatically we also obtain $1 = \operatorname{ind} D^* = \dim D^* = \Ind D^*$.) \medskip \noindent\textbf{2. Dimension of certain dense subsets: }Let $X$ be $T_4$.\\ \resetpr \noindentpr Prove that for every finite open cover ${G_1,\ldots,G_k}$ there exist covers $\mathcal{H} = {H_1,\ldots,H_k}$ and $\mathcal{E} = {E_1,\ldots,E_k}$ such that $\mathcal{H}$ consists of cozero sets, $\mathcal{E}$ consists of zero sets and $H_i\subset E_i\subset G_i$, $i=1,\ldots,k$.\\ \noindentpr Prove that whenever $M\subset X$ is dense and $C^*$-embedded, then for every open cover ${U_1,\ldots,U_k}$ of $M$ consisting of cozero sets in $M$ there exists an open cover ${V_1,\ldots,V_k}$ of $X$ satisfying $V_i\cap M = U_i$ for every $i=1,\ldots,k$.\\ \noindentpr Prove that whenever $M\subset X$ is dense, $C^*$-embedded and normal, then $\dim M\geq \dim X$.\\ (Using Example 1b) above we even obtain $\dim M = \dim X$, in particular we get that $\dim X = \dim \beta X$.) \medskip \noindent\textbf{3. The dimension $\ind$ can be much smaller than $\dim$: }\\ \resetpr \pr Let $X$ be a separable metric space. Prove that there exists a cover ${X_\alpha\colon \alpha<\omega_1}$ of the space $X$ such that $X_\alpha\subset X_\beta$ for $\alpha<\beta$ and $\dim X_\alpha\leq 0$ for every $\alpha<\omega_1$.\\ (\emph{Hint: use that $X\subset [0,1]^\omega$ and thus it suffices to prove the statement for $X = [0,1]^\omega$. You may use that for the interval $[0,1]$ the corresponding decomposition has already been constructed in the Exercises on Connectedness, Example 7a).})\\ \noindentpr Recall that in the Exercises on Connectedness, Example 7, using the interval $[0,1]$ we constructed a normal space $D$ for which in Example 1 above we proved that $\ind D = 0 < 1 = \dim D$. Try to devise how to perform an analogue of this construction and thus construct an example of a normal space $D$ satisfying $\ind D = 0 < \infty = \dim D = \Ind D$.\\ (\emph{Hint: instead of the interval $[0,1]$ in the construction work with $[0,1]^\omega$, to prove that the corresponding constructed set $D$ satisfies $\dim D = \infty$ argue analogously as in Example 1c) above: show that $D$ is $C^*$-embedded in $D^*$ and thus using Example 2 above conclude that $\dim D = \dim D^*\geq \dim [0,1]^\omega = \infty$.\\ Note: if we carried out the construction with $[0,1]^n$, we would obtain a normal space $D$ satisfying $\ind D = 0 < n = \dim D = \Ind D$.}) \medskip \noindent\textbf{4. Small inductive dimension in compact spaces: }\\ \resetpr \noindentpr Prove that for every $T_2$ compact space $X$ it holds that $\ind X = 1$ if and only if $\Ind X = 1$.\\ \noindentpr Let $X$ and $Y$ be $T_2$ compact spaces satisfying $\Ind(X)\leq 1$ and $\Ind(Y)\leq 0$. Show that then $\Ind(X\times Y)\leq 1$.\\ \noindentpr Prove that if a $T_4$ space $X$ is the union of closed subsets $A$ and $B$, then $\ind(X)\leq \ind(A) + \ind (B)$.\\ \noindentpr Let $L_0 = [0,\omega_1)\times [0,1)$ be the linearly ordered space ordered lexicographically and let $L:=L_0\cup {\omega_1}$ be the linearly ordered space where $x\leq \omega_1$ for $x\in L$. Prove that if on $L$ we consider the topology given by the order, then $L$ is a $T_2$ compact space satisfying $\dim L = \ind L = \Ind L = 1$. Note: the space $L$ is called the \emph{long line segment.} \medskip \noindent\textbf{5. The dimension $\dim$ can be smaller than $\ind$: }\\ \resetpr \noindentpr \emph{Construction of two Cantor functions:} On the Cantor discontinuum $2^\omega$ consider the lexicographic order, by the symbol $0\in 2^\omega$ and $1\in 2^\omega$ denote the largest and smallest element in the lexicographic order, for $s\in 2^n$ denote by $x_s\in 2^\omega$ the point such that $x_s|{{1,\ldots,n}} = s$ and $x_s(i)=s(n)$ for $i\geq n$, for $s\in 2^n$ satisfying $s(n-1)=0$ and $s(n)=1$ denote by $s^+\in 2^n$ the sequence given by $s|{{1,\ldots,n-2}} = s^+|{{1,\ldots,n-2}}$, $s^+(n-1)=1$ and $s^+(n)=0$ and consider the countable subset $D:={x_s\colon s\in 2^n,;n\in\en}$. Find continuous surjections $h_i:2^\omega\to [0,1]$ for $i=1,2$ such that they are nonincreasing (i.e. whenever $x