\documentclass{homework651}
\include{hwextras}
\usepackage{graphicx}
\usepackage[all,cmtip]{xy}
\def\net<#1>{\left<#1\right>}

\newcommand{\bbB}{\mathbb{B}}
\doclabel{Math F651: Homework 9}
\docdate{Due: April 5, 2026}
\begin{document}
\begin{problems}
\problem Suppose $\left<x\right>_{\alpha\in A}$ is a net
in $X$ that does not converge to $x\in X$.  Show that
there is an open set $U$ containing $x$ and a subnet
$\left<x_{\alpha_\beta}\right>_{\beta\in B}$ such that
$x_{\alpha_{\beta}}\not\in U$ for all $\beta\in B$.
Hint: For a particular `bad` $U$, 
take $B$ to be the entire subset of $A$ such that
$x_{\beta}\not\in U$ and show that $B$ is directed.
Then show that there is a natural increasing cofinal map
from $B$ to $A$.


\problem Crossley 6.1
Show that the spaces $[0,1]$ and $(0,1)$ are homotopy equivalent
by finding an explicit homotopy equivalence and its inverse 
between the two spaces.

\problem Crossley 6.4

Suppose $f:X\rightarrow S^n$ is continuous and not surjective.
Show that it is homotopic to a constant map.

\problem Crossley 6.5

Show by means of an explicit homotopy
that the map $f:S^1\rightarrow S^1$ given by $f(x,y)=(-x,-y)$
is homotopic to the identity.

\problem Show that a space $X$ is contractible if and only if 
$[X,X]$ consists of a single element.

\problem Lee 7-1


\end{problems}


\end{document}

