\chapter{Topological sequences}
\section{Topological sequences}

%\begin{definition}[Limit point of a sequence]
%	In a topological space $(X,\mathcal{T})$, a \emph{limit point of a sequence} is a point $p$ where all its neighborhoods contain all remaining terms of a sequence. A \emph{convergent sequence} is a sequence with a limit.
%\[  (X,\mathcal{T}) \]
%\[  p \text{ is a limit point of } (x_n)_{n \in \mathbb{N}} \iff \forall V \subseteq X [ V \text{ is a neighborhood of }p \implies \exists N \in \mathbb{N} [  n > N \implies  a_n \in V   ]  ]  \]
%\end{definition}

Some readers may know that convergent sequences of real numbers converge to a single real number, however for general topological spaces, limit points are not necessarily unique.

If one restricts the terms of a sequence to some set, it may still be possible that the limit of the sequence lies \emph{outside} this set. Consider the Euclidean topology on $\mathbb{R}$, the open set $(0,1)$ and the sequence $a_n = \frac{1}{n+1}, n \geq 1$. Though we have $a_n \in (0,1)$, we also have $\lim_{n \to \infty} a_n = 0$, which is out of the set!

The phenomenon where limits can exceed the set their terms are chosen from is interesting indeed; any point that is the limit of some sequence of terms within a set is called a \emph{limit point} of that set.

\section{Nets}
\section{Filters}












\chapter{Advanced results on metric spaces}

Isometries can be interpreted well by means of group theory. Expressing our knowledge group theoretically, the orthogonal linear maps (rotations) $\mathrm{O}(n,\mathbb{R})$, the translation maps $\mathrm{T}(n,\mathbb{R})$  and their composition generate the isometries of $\mathbb{R}^n$. Since taking the product of subgroups with these groups is a semidirect product $\mathrm{O}(n,\mathbb{R}) \rtimes \mathrm{T}(n,\mathbb{R})$, it is a group; we call it the \emph{Euclidean group}.

\begin{definition}[Euclidean group]
Group of all isometries on $\mathbb{R}^n$
$\mathrm{E}(n) = \mathrm{O}(n,\mathbb{R}) \rtimes \mathrm{T}(n,\mathbb{R})$
\end{proposition}


\begin{theorem}[Heine-Cantor theorem]
\[f : X \to Y\]
\[f \text{ is continuous } \land X \text{ is sequentially compact } \implies f \text{ is uniformly continuous}\]
\end{theorem}

\chapter{Baire spaces}

\begin{theorem}[Baire category theorem]
Complete metrizable spaces are Baire spaces
\end{theorem}





\chapter{Separable spaces}

A separable topological space is a topological space such that there exists a countable dense subset.



\chapter{Cauchy spaces}

Topological spaces don't have enough "language" to describe some of the concepts of metric spaces. We may consider extra structure to consider more properties on topological spaces.


\chapter{Uniform spaces}