Lemma 4.4.3. Let $(X, \topo)$ be a topological space, then $U \subset X$ is open if and only if $U \in \cn_{\topo}(x)$ for all $x \in U$.
Proof. Suppose that $U \in \cn_{\topo}(x)$ for all $x \in U$. For each $x \in U$, there exists $V_{x} \in \topo$ with $x \in V_{x} \subset U$. Thus $U = \bigcup_{x \in U}V_{x} \in \topo$.$\square$