Definition:Convergent Filter Basis

Definition

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

Let $\mathcal B$ be a filter basis of a filter $\mathcal F$ on $X$.

Then $\mathcal B$ converges to a point $x \in X$ iff:

$\forall N_x \subseteq X: \exists B \in \mathcal B: B \subseteq N_x$

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

That is, a filter basis is convergent to a point $x$ if every neighborhood of $x$ contains some set of that filter basis.

Sources

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