\documentclass[12pt]{article} \pagestyle{empty} \usepackage{amsfonts} \usepackage{amsmath} \usepackage[letterpaper,margin=.8in]{geometry} \usepackage{fancyhdr} \usepackage{enumitem} \setlist[itemize]{topsep=0pt,before=\leavevmode\vspace{-1.5em}} \setlist[description]{style=nextline} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \def\nrml{\trianglelefteq} \chead{Topology Comprehensive Exam} \lhead{2025 Fall} \pagestyle{fancy} \renewcommand{\headrulewidth}{0pt} \begin{document} \begin{center} {\large\textbf{Topology Comprehensive Exam}}\hfill{Name:$\underline{\hspace{130pt}}$} \end{center} Answer six (6) questions total. On the first page of your work, please write the numbers of the problems that you want graded. On each page please write only on the front side. \begin{center}\begin{tabular}{|l|c|c|c|c|c|c|l|} \hline Problems & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & Total\\ \hline Scores & & & & & & & \\ \hline \end{tabular} \end{center} \noindent \textbf{1.} Given any set $X\neq \emptyset$, prove that there is a smallest topology on $X$ such that $X$ is $T_1$. What is this topology? Prove that your answer works, that is, that it is a topology, that it $T_1$, and that it is the smallest $T_1$ topology on $X$. $\\$ \noindent \textbf{2.} Let $(X,\tau)$ be a topological space. Let $\mathcal{C}\subseteq \tau$ be a collection of sets with the property that given any $U\in \tau$ and any $x\in U$, there exists $C\in \mathcal{C}$, such that $x\in C\subseteq U$. \begin{enumerate}[label=\alph*.] \item Prove that $\mathcal{C}$ is a basis. \item Prove that $\mathcal{C}$ is a basis for the topology $\tau$. \end{enumerate} \noindent \textbf{3. } Let $(X,\tau)$ be a topological space, and let $\mathcal{C}$ be the collection of closed sets. A \emph{filter} on $\mathcal{C}$ is a collection $\mathcal{F}$ of sets from $\mathcal{C}$, such that (1) $\emptyset\not\in \mathcal{F}$, (2) if $C_1,C_2\in \mathcal{F}$, then $C_1\cap C_2\in \mathcal{F}$, and (3) if $C_1\subset C_2$ with $C_1\in \mathcal{F}$ and $C_2\in \mathcal{C}$, then $C_2\in \mathcal{F}$. Show that if each filter on $\mathcal{C}$ has non-empty intersection, then $(X,\tau)$ is compact. $\\$ \noindent \textbf{4. } Let $f:\mathbb{R}\rightarrow \prod_{i \in \mathbb{Z}^+} \mathbb{R}$ be the map given by $f(t)=(t,\frac{1}{2}t,\frac{1}{4}t,\frac{1}{8}t,\frac{1}{16}t...)$. \begin{enumerate}[label=\alph*.] \item Define the box topology on $\prod_{i \in \mathbb{Z}^+} \mathbb{R}$. \item Prove or disprove that $f$ is continuous when $\prod_{i \in \mathbb{Z}^+} \mathbb{R}$ has the \textbf{box topology} and $\mathbb{R}$ has the standard topology. \item Let $d:\R^2\to \R$ be defined by $d(x,y) = \min\{|x-y|,1\}$. Given points $\vec{x},\vec{y}\in \prod_{i \in \mathbb{Z}^+} \mathbb{R}$, let $\rho(\vec{x},\vec{y}) = \sup\{d(x_i,y_i)\mid i\in \mathbb{Z}^+ \}$. You may assume that $d$ and $\rho$ are metrics. The metric topology for $\rho$ is called the \emph{uniform topology} on $ \prod_{i \in \mathbb{Z}^+} \mathbb{R}$. Prove or disprove that $f$ is continuous when $\prod_{i \in \mathbb{Z}^+} \mathbb{R}$ has the \textbf{uniform topology} and $\mathbb{R}$ has the standard topology. \end{enumerate} \noindent \textbf{5.} Consider the product topology $X=\prod_{\alpha\in{\mathbb R}}{\mathbb R}_\alpha$, where each ${\mathbb R}_\alpha$ is a copy of the real line ${\mathbb R}$ using the standard topoolgy, i.e. $X$ is ``the product of ${\mathbb R}$ many real lines''. A point $(x_\alpha)_{\alpha\in{\mathbb R}}$ [i.e. the $\alpha$-component of this point is $x_\alpha\in{\mathbb R}$] can be identified with a function $f: {\mathbb R}\rightarrow {\mathbb R}$, defined by $f(\alpha)=x_\alpha$. Let $f_i$ be a sequence in $X$. Show that $f_i\rightarrow f$ if and only if, viewed as functions, $f_i$ converges to $f$ pointwise [i.e. for any $x\in {\mathbb R}$, we have $\lim_{i\rightarrow\infty}f_i(x)=f(x)$.] $\\$ \newpage \noindent \textbf{6.} For each of the following equivalence relations on topological spaces, give the familiar space to which the quotient space $X /\sim$ is homeomorphic. You DO NOT need to justify your answer. \begin{enumerate}[label=\alph*.] \item Define an equivalence relation on $X=\mathbb{R}$ with the standard topology by setting $$x_1 \sim x_2 \quad \text{if} \quad x_1 - x_2 \in \mathbb{Z}.$$ \item Define an equivalence relation on $X=\mathbb{R}^2$ with the standard product topology by setting $$x_1 \times y_1 \sim x_2 \times y_2 \quad \text{if} \quad x_1 - x_2 \in \mathbb{Z} \quad \text{and} \quad y_1 - y_2 \in \mathbb{Z}.$$ \item Define an equivalence relation on $X=\mathbb{R}^2$ with the standard product topology by setting $$x_1 \times y_1 \sim x_2 \times y_2 \quad \text{if} \quad y_1 - (x_1)^3 = y_2 - (x_2)^3.$$ \item Define an equivalence relation on $X=\mathbb{R}^2$ with the standard product topology by setting $$x_1 \times y_1 \sim x_2 \times y_2 \quad \text{if} \quad (x_1)^2 +(y_1)^2 = (x_2)^2 + (y_2)^2.$$ \item Let $X$ be the topological space $\mathbb{R}^2$ with the dictionary order topology. Define an equivalence relation on $X$ by setting $$x_1 \times y_1 \sim x_2 \times y_2 \quad \text{if} \quad x_1=x_2.$$ \end{enumerate} \noindent \textbf{7. } Prove the following (you may use standard results so long as they do not trivialize the problem): \begin{enumerate}[label=\alph*.] \item Assume $X, Y$ are path-connected, then $X\times Y$ is path-connected. \item Assume $X, Y$ are connected, then $X\times Y$ is connected. \end{enumerate} \noindent \textbf{8. } Assume $X$ is compact, Hausdorff. Assume there is a sequence of compact subsets $A_1\supset A_2\supset A_3 ...$ in $X$. Prove the following (you may use standard results so long as they do not trivialize the problem): \begin{enumerate}[label=\alph*.] \item If for all $n$, $\bigcap_{j=1}^n A_j\neq\emptyset$, prove that $\bigcap_{j=1}^\infty A_j\neq\emptyset$. \item Assume $U$ is an open set so that $\bigcap_{j=1}^\infty A_j\subset U$. Prove that for sufficiently large $n$, $A_n\subset U$. [hint: find an open cover of $X-U$.] \end{enumerate} \noindent \textbf{9. } A topological space $X$ is called \emph{Lindel\"of} if every open cover $\mathcal{U}$ of $X$ has a countable subset $\mathcal{V}\subseteq \mathcal{U}$ such that $X\subseteq \bigcup_{V\in \mathcal{V}} V$. A topological space $X$ is called \emph{Lindelite} if every open cover $\mathcal{U}$ of $X$ has a countable subset $\mathcal{V}\subseteq \mathcal{U}$ such that $X\subseteq \bigcup_{V\in \mathcal{V}} \overline{V}$. \begin{enumerate}[label=\alph*.] \item Define $T_3$ (also known as ``regular''). \item Suppose that $X$ is $T_3$. Prove that if $x\in U$ where $U$ is open, then there exists an open set $W$ such that $x\in W\subset \overline{W}\subset U$. \item Prove that if $X$ is Lindelite and $T_3$, then $X$ is Lindel\"of. \end{enumerate} \noindent \textbf{10. } Let, $X=\{1,2,3,4\}$ and $\tau=\{\emptyset, X, \{1\}, \{1,2\}, \{1,3\}, \{1,2,3\}\}$. Note that $(X,\tau)$ is a topological space. \begin{enumerate}[label=\alph*.] \item Define what it means for a topological space to be normal. \item Prove that $(X,\tau)$ is normal. \item Prove that $A=\{1,2,3\}\subset X$ with the subspace topology is not normal. \end{enumerate} \end{document}