There seems to be many lemmas and propositions in these notes where they skip many proofs(which makes my learning rather hard) because maybe the author thinks it is straightforward, but I don't seem to see why it is the case for instance this lemma on interior points:
Lemma 4.62: $x \in \text{int} (A),$ iff there is an open set $B$, such that $x\in B$ and $x\in B \subset A. $
This lemma does not seem so straightforward to me, can anyone give me a proof to this? I would appreciate the help.
$\mathsf{int}(A)$ can be defined as union of all open subsets of $A$.
As a union of open sets it is open itself, and can be characterized as the "largest" open subset of $A$.
It means actually that: $$B\subseteq A\text{ and }\ B\text{ open }\implies B\subseteq\mathsf{int}(A)\tag1$$
If $x\in\mathsf{int}(A)$ then there is indeed an open set $B$ with $x\in B\subseteq A$ because we can simply take $B=\mathsf{int}(A)$.
If conversely $x\in B\subseteq A$ where $B$ is open then $B\subseteq\mathsf{int}(A)$ as a consequence of $(1)$.