\part{Separation and countability properties}

\chapter{Separation properties}
- T\_0 space (Kolmogorov space)
\begin{definition}
	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}

\begin{definition}
	A \emph{$T_1$ space (Topological 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{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}

- normal space
- Urysohn's lemma

- regular space

- finite frechet spaces are discrete topological spaces
- metrix spaces are Hausdorff spaces

\chapter{Countability properties}

- first countable space
-second countable space

- metric spaces are first countable spaces
- Euclidean spaces are second countable spaces


\part{Nets and Filters}
\part{Topological groups}







- Jordan curve theorem (JCT)
- Urysohn's lemma
