Definition 4.2.11 (Convergence). Let $X$ be a topological space, $\fF \subset 2^{X}$ be a filter with base $\fB \subset 2^{X}$, and $x \in X$, then the following are equivalent:
$\fF \supset \cn(x)$.
For each ultrafilter $\fU \supset \fF$, $\fU \supset \cn(x)$.
There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$, there exists $F \in \fB$ with $F \subset E$.
If the above holds, then $x$ is a limit point of $\fB$, and $\fB$ converges to $x$.
If $A \subset X$ and $\fB \subset 2^{A}$, then $\fB$ converges to $x$ if $\fF(\fB) \supset \bracsn{U \cap A| U \in \cn(x)}$.
Proof. (1) $\Leftrightarrow$ (2): By (2) of Lemma 4.2.10.
(1) $\Rightarrow$ (3): $\cn(x)$ is a fundamental system of neighbourhoods at $x$.
(3) $\Rightarrow$ (1): For any $U \in \cn(x)$, there exists $B \in \cb(x)$ and $F \in \fB$ with $F \subset B \subset U$. In which case, $U \in \fF$.$\square$