Definition 5.2.12 (Accumulation Point).label Let $X$ be a topological space, $\fF \subset 2^{X}$ be a filter with base $\fB$, and $x \in X$, then the following are equivalent:
- (1)
$x \in \bigcap_{E \in \fB}\overline{E}$.
- (2)
$x \in \bigcap_{E \in \fF}\overline{E}$.
- (3)
There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$ and $f \in \fB$, $E \cap F \ne \emptyset$.
- (4)
There exists a filter $\fU \supset \fB$ that converges to $x$.
Proof. (1) $\Rightarrow$ (3): Let $U \in \cn(x)$, then $U \cap E \ne \emptyset$ for all $E \in \fB$.
(3) $\Rightarrow$ (4): By Lemma 5.2.8, there exists a filter $\fU \supset \cb(x) \cup \fB$. Since $\cb(x)$ and $\fB$ are bases for $\cn(x)$ and $\fF$, respectively, $\fU \supset \cn(x) \cup \fF$.
(4) $\Rightarrow$ (1): Let $U \in \cn(x)$, then since $\fU \supset \fB \cup \cn(x)$, $U \cap E \ne \emptyset$ for all $E \in \fB \subset \fF \subset \fU$. Thus $x \in \bigcap_{E \in \fF}\ol{E}$.$\square$