Math and science::Topology

Sequence convergence

Let $X$ be a topological space, let $(x_{n})$ be a sequence in $X$ and let $x_{0} \in X$ . Then $(x_{n})$ converges to $x_{0}$ iff for every neighbourhood $U$ of $x_{0}$ there is an $N \geq 1$ such that for every $n \geq N$ , $x_{n} \in U$ .

In other words, every neighbourhood of the point to which a sequence converges must contain the whole tail end of the sequences starting from some arbitrary point in the sequence.

It is a good exercise to check that this definition carries the expected meaning for metric spaces.

Example

Sequences in Hausdorff spaces

A common demonstration of the importance of Hausdorff spaces:

Let $X$ be a Hausdorff topological space. Then each sequence in $X$ converges to at most one point.

Another good proof exercise.

Multiple convergence

An example where we get multiple convergence:

Sequence convergence