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

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

\doclabel{Math F651: Homework 6}
\docdate{Due: March 1, 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
Although quotient maps must take saturated open sets to open sets
and saturated closed sets to closed sets, a quotient map need not be open or closed.  The point of this exercise is to see an example.

Let $A$ be the set of points $(x,y)$ is $\Reals^2$ with $y=0$ or $x\ge 0$.  Let $\pi(x,y)=x$.
Show that $\pi:A\rightarrow\Reals$ is a quotient map, but that it is neither open nor closed.

\problem
Let $\pi:X\rightarrow Y$ be a quotient map and let $A\subseteq X$
be a saturated closed set or a saturated open set.  Show that $\pi|_A:A\rightarrow \pi(A)$
is a quotient map.

\problem Problem 3-14

\problem Problem 3-16

Your solution should not be longer than a page.  Extra credit for the
shortest correct solution.

\problem
No rigor please on this problem.  Just coherent explanations, and maybe a picture or two.
\begin{subproblems}
\item Define an equivalence class on $\Cplx$ where $z\sim w$ if there is $u\in S^1$ with
$z=wu$.  The quotient space $\Cplx/\sim$ is a familiar topological space.  Name it.
 (``is'' means ``is homeomorphic to, of course''). 
\item Define an equivalence class on $\Reals^2$ where $(x,y)\sim (x+1,-y)$ (along with all the relations then implied by transitivity).  The resulting quotient space is a familiar one.  Name it.
\end{subproblems}


\problem
Let $X=\Reals\times\{0,1\}$ and define an equivalence
relation on $X$ by $(0,0)\sim(0,1)$.  Rigorously show that $X/\sim$
is homeomorphic to the union of the $x$- and $y$-axes in the plane.

\problem Problem 4-1

\problem \exercise{4-4}
\end{problems}
\end{document}