Proposition 1

A filter base $\mathfrak{B}$ on a topological space $X$ converges to $x$ if and only if every set of a fundamental system of neighbourhoods of $x$ contains a set of $\mathfrak{B}$.

In accordance with the terminology introduced in § 1, no. 2 we can state Proposition 1 in the following way: $\mathfrak{B}$ converges to $x$ if and only if there are sets of $\mathfrak{B}$ as near as we please to $x$.

If a filter $\mathfrak{F}$ converges to $x$, then every filter finer than $\mathfrak{F}$ also converges to $x$, by reason of Definition 1. Likewise, if the topology of $X$ is replaced by a coarser topology, the neighbourhood filter of $x$ is replaced by a coarser filter ($\S$ 2, no. 2, Proposition 3), and therefore $\mathfrak{F}$ still converges to $x$ in this new topology.

We can therefore say that the finer the topology, the fewer convergent filters there are in this topology. In particular, in the discrete topology, the only convergent filters are the neighbourhood filters, for these are the trivial ultrafilters on $X$ ($\S$ 6, no. 4).

Let $\Phi$ be a set of filters on $X$, all of which converge to the same point $x$; the neighbourhood filter $\mathfrak{B}(x)$ is coarser than all the filters of $\Phi$, hence also coarser than the intersection $\mathfrak{J}$ of these filters; in other words, $\mathfrak{J}$ also converges to $x$.