\chapter{Separation properties}

The properties we're going to explore arise from a deeper study of topology rather than from inspiration of mathematical analysis.

It's interesting to classify topological spaces by how their points can be separated by open sets. Given that we know how neighborhoods of points should relate to one another, we can prove many interesting results.


\section{Kolmogorov space}


\begin{definition}[Kolmogorov space]
	A \emph{$T_0$ space (Kolmogorov space)} is a topological space such that for every distinct pair of points, at least 1 point in the pair has a neighborhood not containing the other point.
\end{definition}

\section{Fréchet spaces}

\begin{definition}[Fréchet space]
A \emph{$T_1$ space (Fréchet space)} is a topological space such that for every distinct pair of points, both points in the pair has a neighborhood not containing the other point.
\end{definition}

\begin{proposition}
Let $(X,\mathcal{T})$ be a  Fréchet space, then any singleton of $X$ is closed.
\end{proposition}

From this one can prove a modest amount of corollaries, such as that finite frechet spaces are discrete topological spaces.

However, more interesting properties follow a stronger separation property; Hausdorff spaces.


\section{Hausdorff spaces}


Metric spaces have many wonderful properties over any old topological space; we've used our theory our theory of metric spaces to inspire topological spaces, however do metric spaces have any inrinsic separation properties?

Notice that for any pair of points in a metric space, we can find open balls for them that are small enough to be disjoint; every pair of distinct points in a metric space have disjoint neighborhoods then.

We'll now consider the class topological spaces that also have this property, they are known as \emph{Hausdorff spaces}.


\begin{definition}
A \emph{$T_2$ space (Hausdorff space)} is a topological space such that for every distinct pair of points, there exists a pair of neighborhoods of both points which are disjoint.
\end{definition}

Felix Hausdorff originally included this separation property within his own definition of a topological space! It's a nice property indeed, however for the sake of making a more minimalistic and general theory of topology, topological spaces are no longer defined to be Hausdorff spaces.


As discussed, the topologies induced by metric spaces are all Hausdorff spaces, however the converse isn't necessarily true.
\begin{proposition}
Topological spaces induced by metric space are Hausdorff spaces.
\end{proposition}

This hints us to the fact that some of the nice properties that metric spaces offer can be invoked by this separation property rather than the explicit need for distance. If this is true, we'd be able to strengthen many of our theorems on metric spaces to just Hausdorff spaces, let's see what we can do with this goal in mind!



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



\section{Regular spaces}

regular space
normal space
- Urysohn's lemma


\chapter{Countability properties}

\section{First countable spaces}
- first countable space
- metric spaces are first countable spaces
\section{Second countable spaces}
-second countable space
- Euclidean spaces are second countable spaces