Definition:Limit Point/Filter Basis

Definition

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

Let $\FF$ be a filter on the underlying set $S$ of $T$.

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

Definition 1

A point $x \in S$ is called a limit point of $\BB$ if and only if $\FF$ converges on $x$.

$\BB$ is likewise said to converge on $x$.

Definition 2

A point $x \in S$ is called a limit point of $\BB$ if and only if every neighborhood of $x$ contains a set of $\BB$.

Also see

• Equivalence of Definitions of Limit Point of Filter Basis

• Results about limit points of filter bases can be found here.