Definition 1

Let $X$ be a topological space and $\mathfrak{F}$ a filter on $X$. A point $x \in X$ is said to be a limit point (or simply a limit) of $\mathfrak{F}$, if $\mathfrak{F}$ is finer than the neighbourhood filter $\mathfrak{B}(x)$ of $x$; $\mathfrak{F}$ is also said to converge (or to be convergent) to $x$. The point $x$ is said to be a limit of a filter base $\mathfrak{B}$ on $X$, and $\mathfrak{B}$ is said to converge to $x$, if the filter whose base is $\mathfrak{B}$ converges to $x$.

This definition, together with Proposition 4 of § 6, no. 3, gives the following criterion: