Definition:Convergent Filter

Definition
Let $$\left({X, \vartheta}\right)$$ be a topological space.

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

Then $$\mathcal F$$ converges to a point $$x \in X$$ if:
 * $$\forall N_x \subseteq X: N_x \in \mathcal F$$

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

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