Definition:Limit Point/Filter Basis

Definition
Let $\mathcal F$ be a filter on a set $S$.

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

A point $x \in S$ is called a limit point of $\mathcal B$ if $\mathcal F$ converges on $x$.

$\mathcal B$ is likewise said to converge on $x$.

Alternative Definition
A point $x \in S$ is called a limit point of $\mathcal B$ iff every neighborhood of $x$ contains a set of $\mathcal B$.

Also see

 * Equivalent Definitions of Limit Point of Filter Basis