Proposition 2

A filter $\mathfrak{F}$ on a topological space $X$ converges to a point $x$ if and only if every ultrafilter which is finer than $\mathfrak{F}$ converges to $x$.

This is an immediate consequence of the preceding remarks and Proposition 7 of § 6, no. 4.

In general a filter can have several distinct limit points; we shall revert to this question in § 8, no. 1.