\documentclass[minion]{homework651}
\include{hwextras}

\def\calB{\mathcal{B}}
\DeclareMathOperator{\Int}{\mathrm{Int}}

\doclabel{Math F651: Homework 5}
\docdate{Due: February 22, 2023}
%\docauthor{Your Name Here}

\begin{document}

Note: The book has Exercises, which are interspersed among the
prose, and Problems, which appear at the ends of the chapters.
It can be easy to confuse the two.  Exercises are denoted in blue.


\begin{problems}

\problem Suppose $\mathcal B_X$ and $\mathcal B_Y$ are bases for $X$ and $Y$ respectively.
Show that $\mathcal B = \{U\times V: U\in B_X, V\in B_Y\}$ is a basis for $X\times Y$.

\problem Suppose $A\subset X$ and $B\subset Y$.  Use the fact that
the characteristic property of the product topology is characteristic to
show that the subspace topology on $A\times B$ is the the same as its
topology as a product of subspaces.

\problem Show that $(X_1\times X_2)\times X_3$ is homeomorphic to $X_1\times X_2 \times X_3$.
You may not use the words ``open'' or ``closed'' at any point in your proof.  (\textit{Hint:} Use the Characteristic Property, Luke!)


\problem Prove the following.
\begin{subproblems}

\item A projection map from an arbitrary product space is an open map.

\item An arbitrary product of Hausdorff spaces is Hausdorff

\item A countable product of second countable spaces is second countable.

\end{subproblems}

\problem  Problem 3-8

\problem  Problem 3-9

\problem  Problem 3-13 a

\problem \exercise{3.61}

\problem This is a heads up: problem 3-14 will be on the next homework.  Start thinking about it now.  
\end{problems}



\end{document}