\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{$T_0$ space}


\subsection{Topologically indistinguishable points}

In a general topological space, one many have a pair of points that share the exact same neighborhoods. When this is the case we call them \emph{topologically indistinguishable}.

The existence of such topologically indistiguishable points happens to be the pesky reason that limits of topological sequences don't have unique limits, since any topological sequence converging to $p$ also converges to topologically indistinguishable points.




\begin{definition}[Topologically distinguishable pair in $X$]
Let $(X,\mathcal{T}$ be a topological space,  $p,q\in X$, and $p \neq q$. $(p,q)$ is \emph{topologically distinguishable} iff the set of neighborhoods of $p$ doesn't equal the set of neighborhoods of $q$
\end{definition}

Many of the types of separation properties we will study are strong enough to eradicate topological indistinguishability from the space, however these separation properties often have weaker variants designed to hold 'up to indistinguishable points',.


Topologically indistinguishability is a set relation, and furthermore, an equivalence relation; this is great because one can engineer this fact to synthetically modify a topological space to have no topologically indistinguishable points!

\begin{proposition}
Topologically indistinguishability forms an equivalence relation.
\end{proposition}



\subsection{$T_0$ spaces}

We can study spaces with no topologically indistinguishable points (ie. all points are topologically distinguishable); these are known as $T_0$ spaces.

\begin{definition}[$T_0$ space]
A \emph{$T_0$ space (Kolmogorov space)} is a topological space such that all pairs of points are topologically distinguishable.
\end{definition}

Since this is a space where the topologically indistinguishable property never holds, we have $N_p \neq N_q$, so without loss of generality (i.e if it doesn't hold for $p$, it holds for $q$), we have $N_p \subset N_p \cup N_q$, so there exists one neighborhood of $q$ not containing $p$. This leads to the following restatement of the definition of a $T_0$ space.


\begin{proposition}[$T_0$ 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{proposition}




\section{$T_1$ spaces}


\subsection{Separable points}


Even if a pair of points is topologically distinguishable, it is possible to have $N_p \subset N_q$. A stronger separation property forbids even this, meaning that the pair of points have their own respective neighborhoods excluding the other point.


In a general topological space, one many have a pair of points that share the exact same neighborhoods. When this is the case we call them \emph{topologically indistinguishable}.

The existence of such topologically indistiguishable points happens to be the pesky reason that limits of topological sequences don't have unique limits, since any topological sequence converging to $p$ also converges to topologically indistinguishable points.

Many of the types of separation properties of spaces will also look at variants of the property designed to hold 'up to indistinguishable points'. However as we will see, certain separation properties are strong enough to ensure no topologically indistinguishable points.

\begin{definition}[Separable pair in $X$]
Let $(X,\mathcal{T}$ be a topological space,  $p,q\in X$, and $p \neq q$. $(p,q)$ is \emph{topologically indistinguishable} iff for every neighborhood $V_p$, there exists a neighborhood $V_q$ such that $V_p = V_q$.
\end{definition}


\subsection{$T_1$ spaces}


\begin{definition}[$T_1$ space]
A \emph{$T_1$ space (Fréchet space)} is a topological space such that  all pairs of points are separable.
\end{definition}


\begin{definition}[$T_1$ 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}

One can apply this definition to see the following.

\begin{proposition}
$(X,\mathcal{T})$ is a $T_1$ space iff any singleton of $X$ is closed.
\end{proposition}



\subsection{$R_0$ space}

Slightly weaver variant of $T_1$  which only separates p''

\begin{definition}[$R_0$ space]
	A \emph{$R_0$ space (symmetric space)} is a topological space such that for every distinct pair of topologically indistinguishable points, at least 1 point in the pair has a neighborhood not containing the other point.
\end{definition}


\begin{proposition}
$(X,\mathcal{T})$ is a $T_1$ space iff it is $T_0$ and $R_0$
\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}


Recalling $T_1$ spaces, imagine we place the further restriction that not only can we find neighborhoods of both points that exclude the other, but we can find disjoint neighborhoods. This is the idea of a \emph{Hausdorff space}.

\begin{definition}[$T_2$ space]
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}


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 and hence Topological spaces induced by metric spaces are always Hausdorff spaces.


Since topological spaces were made by trying to generalize metric spaces to the closest structure without a metric, 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 definition for a topological space and a more general theory of topology, topological spaces are no longer defined to be strictly 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}


\subsection{$R_1$ space}

Like how $R_0$ spaces are weakened $T_1$ spaces that only hold up to topologically indistinguishable points, one can do something similar for Hausdorff spaces.

\begin{definition}[$R_1$ space]
A \emph{$R_1$ space (Preregular space)} is a topological space such that for every distinct pair of topologically indistinguishable points, both points in the pair has a neighborhood not containing the other point.
\end{definition}

As expected, if all points are topologically distinguishable this just culminates in a Hausdorff space.

\begin{definition}[$T_2$ space]
A Hausdorff space is a topological space that is $R_1$ and $T_0$
\end{definition}




\section{Regular spaces}

regular space
normal space
- Urysohn's lemma


\chapter{Countability properties}

\section{First countable spaces}
- first countable space
countable neighborhood basis



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

- Euclidean spaces are second countable spaces


\section{Lindeloef space}
all open covers have countable subcovers

it is a more general variant of compact spaces, evidently, all compact spaces are Lindeloef spaces.

separable space
there exists a countable set that is dense in the top9logical space

$\mathbb{Q}$ is countable and dense in $\mathbb{R}$ with Euclidean topology, hence Euclidean space is separable.
