\documentclass[a4paper,11pt]{article}

\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage[latin1]{inputenc}
\usepackage[italian]{babel}
\usepackage{latexsym}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bQ}{\mathbb{Q}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\1}{{\rm 1}\!{\rm I}}
\newcommand{\di}{\partial}
\newcommand{\dbar}{\bar{\partial}}
\newcommand{\Qbar}{\bar{\bQ}}
\newtheorem{ex}{Exercise}
\newtheorem{example}{Example}
\newtheorem{thm}{Theorem}
\newtheorem{defi}{Definition}

\begin{document}
\title{Homework 3}
\author{Renzo's math 570}
\date{}

\maketitle

Third homework set. Have fun!

\section{Product Spaces, Connectedness, Compactness}

Remember that the product topology is induced on the cartesian product of topological spaces to be the coarsest topology that makes the projections continuous.



\begin{ex}
Consider the cartesian product of infinitely many topological spaces:
$$
X=\prod_{i \in \bN} X_i
$$
Consider the following two topologies on $X$:
\begin{itemize}
	\item $\tau_1$=\{topology generated by box sets, i.e. an open set is $U=\prod U_i$, with each $U_i$ open in $X_i$ \}.
	\item $\tau_2$=\{topology generated by sets of the form $U=\prod_{i=1}^n U_i\times \prod_{i>n}X_i$\}.
\end{itemize}
\begin{enumerate}
	\item Are these two topologies different? Is one finer than the other?
	\item Is either of these topologies the product topology? Explain why.
\end{enumerate}
\end{ex}

\begin{ex}
Prove that $X\times Y$ is Hausdorff if and only if both $X$ and $Y$ are.
\end{ex}


\begin{ex}
Prove that the projection $\pi_X: X \times Y\rightarrow X$ is an open function (i.e. if $U$ is open, then $\pi_X(U)$ is open). Give an example to show that the projections in general are not closed functions. However show that if $X$ is Hausdorff and $X\times Y $ compact, then the projection to $X$ is a closed function.
\end{ex}

\begin{ex}
What sets are connected topological spaces when they are given the discrete topology? And the stupid topology?

What sets are compact topological spaces when they are given the discrete topology? And the stupid topology?
\end{ex}



\begin{ex}
Prove that $X\times Y$ is path connected if and only if both $X$ and $Y$ are.
\end{ex}

\begin{ex}
Prove that a product space is compact if and only if it is ``compact for box sets covers''(i.e. if any open cover by box sets can be refined to a finite subcover).
\end{ex}

\begin{ex}
Prove that the projective plane is a compact topological space.
\end{ex}

\begin{ex}
Prove the easy direction of Heine Borel. If $X$ is a compact subset in $\bR^n$, then $X$ is closed and bounded.
\end{ex}

\end{document}

