\documentclass[12pt]{article}
\usepackage{amssymb,amsmath,amsthm,amsfonts,amscd,latexsym, dsfont, color, xcolor}
\usepackage{hyperref}
\hypersetup{colorlinks}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\op}[1]{\operatorname{#1}}
\usepackage[margin=.75in]{geometry}
%\setlength{\topmargin}{0in}
\setlength{\textheight}{9in}
\begin{document}
\pagestyle{empty}
{\bf Math 444/539 HW\#1, due Wednesday 9/4/19} \hspace{.5in} \textbf{NAME:} \hspace{4in}
%outlier malcom
\begin{enumerate}
\item Show that a space which is connected and locally path connected is path connected.
\vfill
\eject
\item[] Recall that a subset $A$ of a topological space $X$ is called a \emph{retract} of $X$ if there exists a continuous map $r: X \to A$ (called a retraction) such that $r(a)=a$ for any $a\in A$.
\item Prove that the relation ``is a retract of'' is transitive, i.e. if $A$ is a retract of $B$ and $B$ is a retract of $C$, then $A$ is a retract of $C$.
\vfill
\eject
\item[Defintion:] A subspace $A \subset X$ is called a \textbf{strong deformation retract} of $X$ if there exists a homotopy $F: X \times I \to X$ such that
\[
\begin{array}{lcll}
F(x,0) &=& x & \\
F(x,1)& \in & A & \\
F(a,t) & =& a & \mbox{ for } a\in A \mbox{ and all } t\in I.
\end{array}
\]
The subspace $A$ is merely a \textbf{deformation retract} if the last equation holds only when $t=1$.
\item Show that a deformation retract of a Hausdorff space must be a closed subset.
\vfill
\eject
\item Give an example of a space which is connected but not path connected. Be sure to show that it is in fact connected but not path connected. \\ \emph{Hint: What can you do to the graph $y=\sin\left(\frac{1}{x}\right)$?} {\bf DO NOT GOOGLE THIS.}
\vfill
\eject
\item[*math:] Prove that if $A$ is a retract of a topological space $X$, $r: X \to A$ is a retraction, $i:A \hookrightarrow X$ is inclusion, and $i_*(\pi_1(A))$ is a normal subgroup of $\pi_1(X)$, then $\pi_1(X)$ is the direct product of the subgroup image $i_*$ and kernel $r_*$.
\vfill
\eject
\item[everyone:] How difficult was this assignment? How many hours did you spend on it? Indicate if you are math graduate student or not.
\vfill
\eject
\end{enumerate}
\end{document}