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\begin{document}
\title{Projects}
\author{Renzo's math 570}
\date{}

\maketitle


\section{Compactifications}
Compact spaces are very nice for many different reasons, most of which will only become evident much further along in your mathematical journey.  Given a  non compact space $X$, one would sometimes like
to construct a homeomorphism from $X$ to an open dense set of a compact space $\tilde{X}$. (Informally, this means, adding some points so that $X$ "becomes compact".) $\tilde{X}$ is called a \textbf{compactification} of $X$.

\vspace{0.3cm}
Assume throughout that $X$ is:
\begin{itemize}
	\item Hausdorff.
	\item \textbf{Locally compact:} this simply means that every point of $x$ is contained in a compact neighborhood.
	\end{itemize}

Define the \textbf{one-point compactification} $\tilde{X}$	of $X$:
\begin{description}
	\item[as a set:] $\tilde{X}$ has precisely one more point that the points of $X$. We suggestively call this point \textbf{infinity}.
	$$
	\tilde{X}= X\cup \{\infty\}
	$$
	\item[topology:] we discuss open sets according to whether they contain $\infty$ or not:
	\begin{itemize}
		\item if $\infty \not\in U$, then $U$ is open in $\tilde{X}$  if and only if it is open in $X$.
		\item if $\infty \in U$, then $U$ is open in $\tilde{X}$  if and only if its complement is compact.
	\end{itemize}
\end{description}
	

	\begin{q}
	Show that for any open set  $U$ in $\tilde{X}$, $U\cap X$ is open in $X$.
	\end{q}
 
 \begin{q}
If $X$ is not compact, then prove that:
\begin{itemize}
	\item $X$ is homeomorphic to an open dense set in $\tilde{X}$.	\item$\tilde{X}$ is compact.
	\item$\tilde{X}$ is connected.
	\item$\tilde{X}$ is Hausdorff.
\end{itemize}
What happens if you start with an $X$ that is already compact?
	\end{q}
	
	\begin{q}
	Show that if $X$ and $Y$ are homeomorphic, so are their one point compactifications.
	\end{q}

 
 
 \begin{q}
 Show that the one point compactification of the plane (with euclidean topology) is (homeomorphic to) the sphere.
 \end{q}
 
 \begin{q}
 What is the one point compactification of:
 \begin{itemize}
 	\item an open disc in the plane.
	\item  a closed disc in the plane minus a point.
	\item an open interval of the real line.
	\item a bunch of open intervals of the real line.
 \end{itemize}
 \end{q}

\begin{q}
Find an (or even better, some) example(s) of two spaces $X$ and $Y$ that are NOT homeomorphic, but such that their one point compactifications are homeomorphic.
\end{q}

\begin{q}
Prove that $\Proj$ is compact, and that $\Proj$ is a compactification of the plane. At this point can you decide whether it is a one-point compactification or not?   
 \end{q}
\end{document}
