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

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

\doclabel{Math F651: Homework 4}
\docdate{Due: February 15, 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 Problem 2-23

\problem Problem 3-2

\problem Problem 3-3 (you'll want to read the hugely useful Gluing Lemma first)

\problem \exercise{3.7}

\problem Give a rock solid proof that the cylinder 
$M=\{(x,y,z)\in\Reals^3:x^2+y^2=1\}$
is a $2$-manifold.

\problem Using metric space arguments only, show that
a sequence $\{x_n\}$ in $\Reals^k$ converges to a limit $x$
if and only if each projection sequence $\{\pi_j(x_n)\}$ 
converges to $\pi_j(x)$, $1\le j\le k$.

\problem Problem 3-6 (wait until after Friday to start this)

\problem (Thanks Torin for musing this question out loud)\

Let
$X$ and $Y$ be topological spaces such that every
\[
f:X\to Y
\]
is continuous.  Show that either $X$ is discrete or $Y$ is indiscrete.

\end{problems}

\end{document}
