Definition:Convergent Filter Basis

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Let $\left({X, \tau}\right)$ 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 $x \in X$ if and only if:

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

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.