Definition 4.5.1 (Interior). Let $X$ be a topological space, $A \subset X$, $x \in X$, and $\fB \subset \cn(x)$ be a fundamental system of neighbourhoods, then the following are equivalent:
$A \in \cn(x)$.
There exists $U \in \fB$ with $U \subset A$.
There exists $U \in \cn(x)$ with $U \subset A$.
The set of all points satisfying the above is the interior $A^{o}$ of $A$, which is the largest open set contained in $A$.
Proof. $(1) \Rightarrow (2)$: Since $\fB$ is a fundamental system of neighbourhoods, there exists $U \in \fB$ with $U \subset A$.
$(2) \Rightarrow (3)$: $\fB \subset \cn(x)$.
$(3) \Rightarrow (1)$: By (F1) of $\cn(x)$, $A \in \cn(x)$.
Let $U \subset A$ be open, then $U \in \cn(x)$ for all $x \in U$ by Lemma 4.4.3. By (3), $U \subset A^{o}$, so $A^{o}$ contains every open subset of $A$. On the other hand, for any $x \in A^{o}$, (2) implies that there exists $U \in \cn^{o}(x)$ with $U \subset A$. In which case, $U \subset A^{o}$, and $A^{o} \in \cn(x)$. Therefore $A^{o}$ itself is open by Lemma 4.4.3.$\square$