\chapter{Compact spaces}


We will actually visit 2 different definitions of compactness; compact spaces and sequentially compact spaces. Historically the latter first came to mind in the context of real analysis as a special space with the guaranteed existence of a convergent subsequence for any sequence, the former arose as an equivalent definition in nice enough spaces, however was found to be a different concept in its own right for general topological spaces.

We'll stary by considering sequentially compact spaces, then compact spaces, establish their equivalence in metric spaces and explore their differences in general topological spaces.


\section{Sequentially Compact space}



Vague introduction to Sequential compactness, shortcomings
This definition is much more intuitive; no sequence

We've studied sequentially compact metric spaces, however we now consider this definition in a general topological sense to understand how it differs from general compactness in the last section.

\begin{definition}[Sequentially compact space]
\end{definition}

\begin{definition}[Sequentially compact set]
\end{definition}



Recall the Heine-Borel theorem, since we know that in metric spaces sequential compactness and compactness are equivalent, the Heine-Borel theorem implies the Bolzano-Weierstrass theorem!

\begin{theorem}[Bolzano-Weierstrass theorem]
$S$ is sequentially compact in $\mathbb{R}^n$ with Euclidean topology iff $S$ is closed and bounded.
\end{theorem}


\subsection{Sequentially compact metric spaces}

\begin{proposition}
Let X be a metric space, then X is sequentially compact iff any open cover of $X$ has a finite subcover.
\end{proposition}


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







\section{Compact space}



We have studied the notion of sequentially compact spaces in the context of metric spaces; the whole idea is that such spaces cramp all sequences in sothat they must have some convergent subsequence.

We also found a set-theoretic condition that can define sequential compactness in metric spaces, however this set-theoretic condition does not describe sequential compactness for a general topological space, because the notions of sequential limits and limit points coincide within metric (and Hausdorff) spaces, but not necessarily any space.

Despite this, we can study the properties of this set-theoretic condition as its own definition and understand how it distinguishes topological spaces and how it relates to sequential compactness.



%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.


%Another property that can be attributed to spaces and be used to distinguish them is whether they are compact.

%In the Euclidean topology, the sets $(0,1)$ and $[0,1]$ contain none and all of their limit points respectively, but consider the subspaces they induce; they aren't homeomorphic. The same goes for comparing any closed ball and open ball of $\mathbb{R}^n$


%The notion of compactness with its formal definition is not as natural as connectedness, so we merely present its definion and build intuition later.

\begin{definition}[Compact space]
Let $(X,\mathcal{T})$ be a topological space, then $(X,\mathcal{T})$ is a \emph{compact space} iff every open cover of $X$ has a finite subcover of $X$.
\[X = \cup_{i \in I} U_i\]
\[X = \cup_{i=1}^{k} U_{n_i}\]
\end{definition}

Since $X$ is bein interpreted as an entire space (no elements outside of $X$), we use equality.
Similarly to how we extended connected spaces to generic connected sets, we can do the same with compactness.

\begin{definition}[Compact set]
Let $(X,\mathcal{T})$ be a topological space, then $S$ is a \emph{compact set of $X$} iff the topological subspace induced by $S$ is a compact space.
\end{definition}


\begin{definition}
Let $(X,\mathcal{T})$ be a compact space, a closed set $F$ is a compact set.
\end{definition}


\begin{theorem}
Let $(X,\mathcal{T})$ be a topological space, then $S$ is a compact set of $X$ iff every open cover of $S$ has a finite subcover of $S$.
\[X \subseteq \cup_{i \in I} U_i\]
\[X \subseteq \cup_{i=1}^{k} U_{n_i}\]
\end{theorem}

compactness preserved under homoemoprhism
compactness preserved under continuous functions
closed intervals are compact in Euclidean space
compactness need only open cover by basic sets

\begin{theorem}[Tychonoff's theorem]
$\prod_{i\in I}S_i$ is compact in product topological space iff each $S_i$ is compact.
\end{theorem}


\subsection{Heine-Borel theorem}

Since sequential compactness coincides with compactness for metric spaces and $\mathbb{R}^n$ is a metric space, the Bolzano-Weierstrass theorem immediately implies the \emph{Heine-Borel theorem}.

\begin{theorem}[Heine-Borel theorem]
$S$ is compact in $\mathbb{R}^n$ with Euclidean topology iff $S$ is closed and bounded.
\end{theorem}




\section{Compactifications}
- One-point compactification
