\documentclass[letterpaper,12pt]{article}

\usepackage{amsfonts,amssymb,amsmath,amsthm,txfonts,pxfonts,mathrsfs,fullpage,enumitem}
\usepackage{xcolor}

\pagestyle{empty}
\newcommand{\RR}{{\mathbb R}}
\begin{document} 

\begin{center}\textbf{Problem Set 6 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 3/21/16} \end{center}

\noindent You are expected to do the following problems to a high standard (i.e., at least well enough to be published in a textbook) for full credit.

\begin{enumerate}

\item {[Baker 5.2.6]} Let $X$ be a connected topological space with $a,b \in X$ and let $f:X \to \mathbb{R}$ be a continuous function (where $\mathbb{R}$ has the usual topology).  Prove that if $y$ is any number between $f(a)$ and $f(b)$, then there is an element $x \in X$ such that $f(x)=y$.  (This is a slightly stronger version of the Intermediate Value Theorem.)

\item {[Baker 5.2.7]} Complete the proof of Theorem 5.2.1.

\item {[Baker 5.2.13]} Determine if The Intermediate Value Theorem holds if the $\mathscr{U}$-relative topology on $[a,b]$ is replaced by the $\mathscr{H}$-relative topology.

\item {[Baker 5.2.21]} Let $X = \{a,b,c\}$ and $\mathscr{T} = \{X,\varnothing, \{a\},\{b\},\{a,b\} \}$.  Find all cut points of $(X,\mathscr{T})$.

\item {[Baker 5.2.22]} Let $X$ and $Y$ be topological spaces and let $f:X \to Y$ be a homeomorphism.  If $x$ is a cut point of $X$, then $f(x)$ is a cut point of $Y$.

\item {[Baker 5.2.23]} In the previous statement, determine if the requirement that $f$ be a homeomorphism can be replaced by the condition that $f$ be a continuous function.

\end{enumerate}

\end{document}