\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}