\documentclass[minion]{homework}
\usepackage{cmacros}

\doclabel{Math F651: Homework 3}
\docdate{Due: February 9, 2011}

\begin{document}
\begin{aproblems}

\hproblem Munkres 18.1

\hproblem Munkres 18.3 

\hproblem Munkres 18.7

\hproblem Let $Y_d$ be a topological space with the discrete topology.  What are 
the continuous maps are from $\Reals$ to $X_d$?  From $\Reals-\{0\}$ to $X_d$?  Prove your
claims.

\hproblem   If $f:X\ra Y$ is a map between topological spaces, we say
that $f$ is {\bf open} if $f(U)$ is open for every open set in $X$.
Suppose $f:X\ra Y$ is an open continuous map.
\begin{enumerate}
\listlabel{\alph{\thelistlabel})}
\item Show that $f$ is a homeomorphism if and only if $f$ is bijective.
\item Show that if $f$ is surjective, and if $\cal B$ is a basis for $X$,
then the collection $\{f(B): B\in{\cal B}\}$ is a basis for $Y$.
\item Find a map from a subset of $\Reals^2$ to a subset of $\Reals^2$ that is open but not 
continuous.
\end{enumerate}

\hproblem Let $f:X\ra Y$ be continuous and let $\mathcal B$ be a basis for $X$.  Let $f(\mathcal B)$
denote the collection $\{f(B):B\in \mathcal B\}$.  If $f$ is surjective and open, prove that 
$f(\mathcal B)$ is a basis for $Y$.

\hproblem Munkres 18.9

\end{aproblems}

\end{document}
