\documentclass[11pt]{article}
\usepackage{amsmath,amsthm,amssymb,fancyhdr,enumitem,newpxtext,newpxmath,tikz,tikz-cd}
\usepackage[top=1.5in]{geometry}


\theoremstyle{definition}
\newtheorem{problem}{}
\newcommand{\bp}{\begin{problem}}
\newcommand{\ep}{\end{problem}\bigskip}


\newcommand{\ds}{\displaystyle}


\begin{document}
\pagestyle{fancy}
\fancyfoot[R,C,L]{}
\fancyhead[R]{\large \textbf{Fall 2022}}
\fancyhead[L]{\large \textbf{Topology Homework 5}}

\bp \textbf{[Problem 6 from Spring 2021]}

\bigskip

\noindent Suppose that the following is a pullback diagram in $\mathsf{TOP}$:
	\[ 
\begin{tikzcd}
    F \arrow[r] \arrow[d,"P"']
    \arrow[dr, phantom, "\lrcorner", very near start]
    & E \arrow[d,"p"] \\
    C \arrow[r]
    & B
\end{tikzcd}
\]
Prove that if $p$ is monic then $P$ is monic.


\ep

\bp  \textbf{[Problem 3 from Fall 2022]}
\bigskip

\noindent Let $X$ and $Y$ be spaces and let $f:X \to Y$ be a function.  Prove that $f$ is a continuous injection if and only if the following diagram is a pullback square:
    \[ 
    \begin{tikzcd}
    X \arrow[r,"id"] \arrow[d,"id"'] & X \arrow[d,"f"]\\
    X \arrow[r,"f"'] &Y
    \end{tikzcd}
\]
\ep


\bp \textbf{[Problem 3 from Fall 2016]}

\bigskip

\noindent Prove that the following two descriptions of the torus are homeomorphic: 

\begin{enumerate}
\item $S^1 \times S^1$
\item A square with opposite sides identified as shown.
\end{enumerate}

\[
\begin{tikzpicture}[scale=1]
\draw [fill=black!20!,very thick] (0,0) rectangle (2,2);
\draw [black,fill] (0,0) circle [radius=0.07];
\draw [black,fill] (2,0) circle [radius=0.07];
\draw [black,fill] (0,2) circle [radius=0.07];
\draw [black,fill] (2,2) circle [radius=0.07];
\tikzset{arrows={-Triangle[angle=45:12pt,black,fill=black]}}
\draw [very thick] (0,0) -- (1.2,0);
\draw [very thick] (0,2) -- (1.2,2);
\tikzset{
arrows={-Triangle[angle=45:14pt,black,fill=white]}}
\draw [very thick] (0,0) -- (0,1.25);
\draw [very thick] (2,0) -- (2,1.25);
\end{tikzpicture}
 \]

\ep

\bp 
\textbf{[Problem 2 from Spring 2016]}

\bigskip

\noindent Suppose $X$ is contractible and $Y$ is any space.  Prove or disprove $X\times Y$ is homotopy equivalent to $Y$.

\ep

\pagebreak

\bp \textbf{[Problem 4 from Spring 2012 $\approx$ problem 4 from December 2020]}

\bigskip

\noindent Suppose $A$ is a subspace of $X$ and $f:A \to Y$.  We say \emph{$f$ can be extended to $X$} if and only if there exists a map $g:X \to Y$ with $g=f$ on $A$.  

\begin{enumerate}[label=(\alph*)] 

\item Prove that if $A$ is a dense subspace of $X$ and $Y$ is Hausdorff, then $f:A \to Y$ can be extended to $X$ in at most one way.
\item Give an example of spaces $X, Y$, a dense subset $A\subseteq X$, and a map $f:A \to X$ that cannot be extended to $X$.  

\item Give an example of spaces $X, Y$, a dense subset $A\subseteq X$, and a map $f:A \to X$ that can be extended to $X$ in more than one way.
\end{enumerate}

\ep

\bp 
\textbf{[Problem 1 from Fall 2021 and Problem 2 on Spring 2021]}

\bigskip

\noindent Prove or disprove:

\begin{enumerate}[label=(\alph*)] 

\item \textbf{[Problem 1a from Fall 2021]}  Compact is a homotopy invariant.
\item \textbf{[Problem 1b from Fall 2021]}  Connected is a homotopy invariant.
\item \textbf{[Problem 2 on Spring 2021]}  Hausdorff is a homotopy invariant.

\end{enumerate}
\ep

\end{document}
