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.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Filters