\documentclass[10pt]{article}
\usepackage{latexsym,amsfonts,amsmath}
\usepackage{fontenc}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\x}{\underline{x}}
\newcommand{\y}{\underline{y}}
\newcommand{\T}{\mathcal{T}}


\begin{document}
\thispagestyle{empty}

\begin{center}
{\Large Intro to Topology -- Hw6}
\end{center}

\bigskip\noindent
\begin{enumerate}




\item In class we  showed claim 1: if $K$ is compact in a Hausdorff space $X$, then
for any $x\in X\setminus K$ there exist open sets $U$ and $V$ 
with $U\cap V=\emptyset$ such that $K\subset U$ and $x\in V$.

That is: in a Hausdorff space, a compact set can be separated from a point. 

As a "next level", now show claim 2: if $X$ is Hausdorff, then for any two compact, disjoint sets $K$ and $M$ in $X$
 there exist disjoint open sets $U$ and $V$ such that $K\subset U$ and
$M\subset V$.

That is: in a Hausdorff space, two disjoint compact sets can be separated by disjoint open sets. 

\smallskip\noindent{\it
 No need to start from scratch, but you can use and modify the proof of claim 1. See your class notes and the posted notes for a proof. Note that in class the first proof had to be corrected.}

\item We also showed that the quotient
$X=[0,1]/0\sim 1$ is homeomorphic to $S^1$.

Generalize that proof to show that $$S^1\times S^1=\{(u,v,x,y)\, | u^2+v^2=1,\, 
x^2+y^2=1\}\ \subset\ \R^4$$ is homeomorphic to the torus i.e. the identification space
$\TT=I^2/\sim$, where $I^2=[0,1]\times [0,1]$ and the equivalence 
relation
$\sim$ is $(0,y)\sim (1,y)$ and $(x,0)\sim (x,1)$.



\item Show that if $(X,d)$ is a metric space and $X$ has countable many points (but more than 2 and possibly countably infinite) and $\tau_d$ is the topology generated by $d$, then $(X, \tau_d)$ must be disconnected.


\item
Are the following subsets of $\R^2$ path-connected? If not path-connected, are they connected?
$$ A=\{ (x,y)\, |\, x,y\in\Q\}$$
$$ B=\{ (x,y)\, |\, x\in\Q\mbox{ or }y\in\Q\}$$


\item[Extra Credit]

Is the following set path-connected, connected or neither?
$$ C=\{ (x,y)\, |\, x,y\in\Q\mbox{ or }x,y\in\R\setminus \Q\}$$

{\it You may find this difficult, I really suggest that you work on it in groups.
However, please write up your solutions on your own.}

\end{enumerate}

\end{document}
