Definition:Limit Point/Filter Basis

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

A point $x \in X$ 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
Let $\mathcal F$ be a filter basis of a filter $\mathcal F$ on $X$.

A point $x \in X$ 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