\documentclass[12pt,a4paper,leqno,twoside,report]{amsart}

\textwidth=154mm
\hoffset=-14mm
\usepackage{latexsym}
\usepackage{verbatim}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
%\usepackage[czech]{babel}
\usepackage[utf8]{inputenc}


\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ap}[thm]{Appendix}
\newtheorem{con}[thm]{Conjecture}

\theoremstyle{definition}
\newtheorem{Def}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{prob}[thm]{Problem}
\newtheorem{ex}[thm]{Exercise}

\def\N{\mathbb N}
\def\Z{\mathbb{Z}}
\def\Z2{\mathbb Z_2}
\def\R{\mathbb R}
\def\RP{\mathbb RP}

\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\S{\mathcal{S}}
\def\T{\mathcal{T}}
\def\L{\Lambda}
\def\d{\delta}

\def\df{\partial_\tau ^{F \ot B}}
\def\dg{\partial_\tau ^{G \ot B}}
\def\bo{\partial} %hranice

\def\bD{\bar{\Delta}}
\def\bW{\bar{W}}
\def\t{\tau}
\def\ot{\otimes}
\def\oth{\mathbin{{\otimes}_\t}}
\def\dif{\partial ^{F \ot B}}
\def\dig{\partial ^{G \ot B}}

\def\r{\rho}


\def\cb{C_* (B)}
\def\ce{C_* (E)}
\def\cg{C_* (G)}
\def\cf{C_* (F)}


\def\ep{\varepsilon}
\def\di{\partial}
\def\co{\circ}
\def\lra{\Leftrightarrow}
\def\ra{\Rightarrow}
\def\la{\Leftarrow}
\input xy
\xyoption{all}

\begin{document}


\title{Homework 1}

\maketitle

Obligatory exercises are 1,2 and 3. 

\bigskip 

\begin{ex}
Show that $SS^{k} = S^{k+1}$. \emph{Hint:} Consider the map 
$f\colon S^{k} \times I \to S^{k+1}$, where
\[
f(x,t) = ((\sqrt{1-(2t-1)^2})x, 2t-1)
\]
\end{ex}

\bigskip

\begin{ex}
Let $f\colon X \to Y$. Consider the mapping cylinder $M_f$.
\begin{enumerate}
\item Prove that the inclusion $\iota_X\colon X\hookrightarrow M_f$  is a 
cofibration.
\item Show that  this gives a possibility to factor every map 
$f\colon X\to Y$ as
$$f=r\circ\iota_X$$ 
where $\iota_X$ is a cofibration and $r\colon M_f\to Y$ is a homotopy
equivalence.
\end{enumerate}
\end{ex}

\bigskip

\begin{ex}
Prove: If $X$ is a Hausdorff space, then its diagonal 
$$\Delta=\{(x,x)\in X\times X\}$$ 
is a closed subspace of $X\times X$.
\end{ex}

\bigskip

\begin{ex}   
Let $X$ be a Hausdorff space and $A\subseteq X$ be its retract. 
Prove that $A$ is closed.  \emph{Hint:} Use the previous exercise and the map
$X\to X\times X\colon x\mapsto (x,r(x))$ where $r\colon X\to A$ is a
retraction.
\end{ex}

\bigskip

\begin{ex} 
The consequence of the previous exercise is: If $X$ is Hausdorff
and $A\hookrightarrow X$ is a cofibration, then $A$ is closed.  \emph{Hint:}
$X\times I$ is Hausdorff.
\end{ex}

\end{document}