\part{Compact spaces}

\chapter{Compact space (metric space)}
Let's consider sequences with elements drawn from some set $U$.
Real analysis tells us that closed and bounded intervals 

As with most of this book, we introduce the metric space version first and generalize to topological spaces.
However, as we've seen, limit points of a set and limits of a topological sequence are related, but distinct in general topological spaces (they are the same in metric spaces, however).

This leads to 2 philosophies of what 'compact' means; in the sense of limit points of a set, or in the sense of sequential limits.

\chapter{Compact space (topological space)}
- compact iff every open cover has finite subcover


Students of real analysis are familiar with the following
\begin{theorem}[Bolzano-Weierstrass theorem]
\end{theorem}

This result is can be generalized to any dimension in the following theorem.

\begin{theorem}[Heine-Borel theorem]
\end{theorem}


\begin{lemma}[Lebesgue's number lemma]
\end{lemma}

\chapter{Compactifications}
- One-point compactification
