Topological Sequence

Definition (Topological Sequence)

If $(x_n{)}_{n\in \mathbb{N}}$ is a sequence in $X$, then we say $x_n\to x$ (converges to $x$) if for any open set $U$ where $x\in U$, then $\exists N$ such that $\forall n\ge N,x_n\in U$.

We say $x$ is a limit of the sequence.

In a Topological Space, a sequence can have non-unique limits. Consider $\mathbb{R}$ with a double origin and we took $x_n=1/n$, then we have two different limits of the sequence.

This is different from a Metric Space where a sequence can only have $1$, unique limit point.

Example 1

If $X$ has an discrete topology, then every sequence $\{x_n\}$ converges to every point.