\documentclass[minion]{homework651}
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\doclabel{Math F651: Homework 9}
\docdate{Due: March 26, 2025}

\begin{document}
Several of the problems on this assignment are repeats.  Use nets!
\begin{problems}

\problem Show that a topological space in $X$ is Hausdorff if and only if every 
convergent net in $X$ has exactly one limit.

\problem Consider the product space $X\times Y$.  Find (and prove) a condition
in terms of coordinate functions that characterizes convergence of nets in the product.
Does your condition also work for an arbitrary product?

\problem Show that a space $X$ is Hausdorff if and only if the diagonal in $X\times X$ is closed.

\problem Let $G$ be a topological group and let $H$ be a subgroup. Show that 
$\overline{H}$ is a subgroup.

\problem Suppose $X$ is a space and $Y$ is compact and Hausdorff.
Show that a function $f:X\rightarrow Y$ is continuous if and only if
the graph of $f$ is closed.

\problem Show that the homeomorphism group of a connected
manifold acts transitively.  In other words, show that if $M$ is a 
connected manifold, then for any two points $p$ and $q$ in $M$
there is a homeomorphism $\psi:M\ra M$ such that $\psi(p)=q$.

\end{problems}

\end{document}

