\documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{enumerate} \usepackage{ifthen} \usepackage{graphicx} \usepackage{amssymb} % needs graphicx package. This blows-up math symbols \usepackage{lscape} \setlength{\unitlength}{0.1in} \setlength{\oddsidemargin}{-0.50in} \setlength{\evensidemargin}{-0.50in} \setlength{\topmargin}{-0.50in} \setlength{\headheight}{0.0in} \setlength{\headsep}{0.0in} \setlength{\topskip}{0.0in} \setlength{\textheight}{10in} \setlength{\textwidth}{7.5in} \newcommand{\comp}{ \,{\scriptstyle \stackrel{\circ}{}}\, } \newcommand{\nullset}{\mathrm{O}\!\!\!\!\big/\,} \newcommand{\divides}{\,\Big|\,} \newcommand{\myspace}{\mbox{\parbox[c][0.35in]{0.35in}{\hfill}}} \newcommand{\myspacea}{\mbox{\parbox[c][0.35in]{0.65in}{\hfill}}} \newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}} \newcommand{\force}{{$\mbox{\;}$}} \newcommand{\bigtimes}[1]{ \mbox{\raisebox{-5pt}{$\stackrel{\mbox{\resizebox{0.2in}{!}{$\times$}}}{\mbox{\resizebox{0.15in}{!}{$#1$}}}$}}} \begin{document} \pagestyle{empty} \noindent \parbox{1.5in}{\bf Math 4710/5710} \hfill {\Large \bf In the Midst-term Exam --- In Class} \hfill \parbox{1.5in}{\bf \hfill November $14^{\mathrm{th}}$, 2014} \vspace{0.1in} \noindent {\bf Standard assumptions:} $X$ is a topological space. Subsets are given the subspace topology. Products are given the product topology. \vspace{0.3in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \vspace{0.3in} \noindent {\bf\large 1. (12 points)} An open question. \begin{enumerate}[(a)] \item Using the standard basis for $\mathbb{R}_\ell$ (real numbers with lower limit topology), show $I=(0,1)$ is open. \vfill \item The interval $J=(0,1]$ is not open in $\mathbb{R}$. Why? \vfill \item The interval $J=(0,1]$ is open in $X=(-\infty,1]$. Why? \vfill \end{enumerate} \noindent {\bf\large 2. (10 points)} Let $\mathcal{T} = \{ U \subseteq X \;|\; U \mbox{ is finite.} \} \cup \{ X \}$. \begin{enumerate}[(a)] \item Explain why $\mathcal{T}$ is not a topology for $X$. \vfill \item Can we restrict $X$ so that $\mathcal{T}$ is a topology for $X$? \vfill %\item Suppose that we find the smallest $\mathcal{T}'$ such that $\mathcal{T} \subseteq \mathcal{T}'$ and $(X,\mathcal{T}')$ is a topological space. % What can we say about this space? %\vfill \end{enumerate} \newpage \noindent {\bf\large 3. (9 points)} Let $A \subseteq X$. Suppose that $a_n \in A$ for all $n=1,2,3,\dots$ and $a_n \to a$. Show that $a \in \bar{A}$. \vfill \noindent {\bf\large 4. (13 points)} Let $\displaystyle f : A \to \prod\limits_{j \in J} X_j$ ($A$ and $X_j$'s are topological spaces). For each $a \in A$, we have $f(a) = (f_j(a))_{j \in J}$ for some functions $f_j:A \to X_j$. \begin{enumerate}[(a)] \item Suppose $f$ is continuous. Show that $f_j$ is continuous for all $j \in J$. \vfill \item Suppose that $f_j$ is continuous for all $j \in J$. Show that $f$ is continuous. \vfill \item Let $f: \mathbb{R} \to \mathbb{R}^\omega$ be defined by $f(t)=(t,t,t,\dots)$. Explain why $f$ is continuous if $\mathbb{R}^\omega$ is given the product topology but not continuous if given the box topology. \vfill \end{enumerate} \newpage \noindent {\bf\large 5. (9 points)} Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. \begin{enumerate}[(a)] \item Using the language of metric spaces, give a careful definition of what it means for $f:X \to Y$ to be continuous at $x \in X$. \vspace{1in} \item Using the language of topological spaces, give a careful definition of what it means for $f:X \to Y$ to be continuous at $x \in X$ (this should work if $X$ and $Y$ are just topological spaces and not necessarily metric spaces). \vspace{1in} \item Briefly explain why these concepts are equivalent. \vfill \end{enumerate} \noindent {\bf\large 6. (7 points)} Basic continuity. $X$, $Y$, and $Z$ are topological spaces. Suppose $f:X \to Y$ and $g:Y \to Z$ are continuous. Show $g \circ f$ is continuous. \vfill \noindent {\bf\large 7. (10 points)} Let $f:X \to Y$ be a continuous map between topological spaces $X$ and $Y$ where $X$ is connected. Show $f(X)$ is connected. \vfill \newpage \noindent {\bf\large 8. (10 points)} Not connected. A statement or a hint? \begin{enumerate}[(a)] \item Show $X = [0,1] \cup [2,3]$ is not connected. \vspace{1in} %\item Show $R_\ell$ is not connected. %\vspace{1in} \item Explain why $\mathbb{R} \not\cong S^1$ (the unit circle). \vfill \end{enumerate} \noindent {\bf\large 9. (10 points)} Compact and not compact. \begin{enumerate}[(a)] \item Let $X$ have the finite complement topology. Show $X$ is compact. \vfill \item Show the interval $I=(0,1) \subset \mathbb{R}$ is not compact. \vfill \end{enumerate} \noindent {\bf\large 10. (10 points)} Let $A \subset X$ where $X$ is Hausdorff and $A$ is compact. Suppose that $x_0 \not\in A$. Show that there exists open sets $U$ and $V$ such that $U$ and $V$ are disjoint, $x_0 \in U$ and $A \subseteq V$. \vfill \vfill %\noindent {\bf\large 11. (points)} Let $f: X \to Y$ be a continuous bijection. Suppose that $X$ is compact and $Y$ is Hausdorff. Show that $f$ is a homeomorphism. %\vfill \newpage \noindent \parbox{1.5in}{\bf Math 4710/5710} \hfill {\Large \bf In the Midst-term Exam --- Take Home} \hfill \parbox{1.5in}{\bf \hfill November $14^{\mathrm{th}}$, 2014} \vspace{0.1in} \force \hfill {\large {\bf Due:} Tuesday, November $18^{\mathrm{th}}$, 2014 at 5pm. } \hfill \force \vspace{0.3in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \vspace{0.3in} The following are True/Possible/False questions. If the statement is always true, state ``{\sc True}'' and then prove the statement. If the statement is never true, state ``{\sc False}'' and then prove the statement cannot ever hold. If the state is sometimes true and sometimes false, state ``{\sc Possible}'' and then give an example of it holding and an example of it failing to hold. \vspace{0.1in} Each problem is worth {\bf 10 points}. Undergrads should complete {\bf 5} out of {\bf 7} (you may do them all for potential extra credit). Grad students must complete all of these. \vspace{0.1in} \begin{enumerate} \item Let $\mathcal{T}$ be a topology for the set $\mathbb{Z} = \{ \dots, -2,-1,0,1,2,\dots \}$. \vspace{0.1in} Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $|\mathcal{T}|=5$? \vspace{0.1in} \item Let $X$ be a metrizable topological space. \vspace{0.1in} Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $X$ has a non-metrizable subspace? \vspace{0.1in} \item $X$ is a path connected topological space. \vspace{0.1in} Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $X$ is still path connected when given a coarser topology? \vspace{0.1in} \item Let $X$ be a non-empty simply ordered set. Give $X$ the order topology. Let $a,b \in X$ with $a