Definition:Convergent Filter

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Let $\left({S, \tau}\right)$ be a topological space.

Let $\mathcal F$ be a filter on $S$.

Then $\mathcal F$ converges to a point $x \in S$ if and only if:

$\forall N_x \subseteq S: N_x \in \mathcal F$

where $N_x$ is a neighborhood of $x$.

That is, a filter converges to a point $x$ if and only if every neighborhood of $x$ is an element of that filter.

If there is a point $x \in S$ such that $\mathcal F$ converges to $x$, then $\mathcal F$ is convergent.