Definition:Convergent Filter Basis

Definition

Let $\struct {S, \tau}$ be a topological space.

Let $\BB$ be a filter basis of a filter $\FF$ on $S$.

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

$\forall N_x \subseteq S: \exists B \in \BB: B \subseteq N_x$

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

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Filters