accumulation points of intersection of two open sets

Question

Let $A$ and $B$ be open subsets of $\mathbb{R}^2$.

If $0 \in A$ and $0 \in \mbox{acc}(B)$, then show that

$ 0 \in \mbox{acc}(A \cap B),$

where acc$(B)$ is the set of accumulation points of $B$.

Answer 1

Let $O$ be any open neighbourhood of $0$. Then $O \cap A$ is also an open neighbourhood of $0$ so contains infinitely many points of $B$, which are points of $A \cap B$, and so a fortiori does $O$. So $0$ is an accumulation point of $A \cap B$.

In one line $$\aleph_0 \le |(O \cap A) \cap B|= |(O \cap (A \cap B)|$$

No specifics of metrics or the plane are needed.

Answer 2

Note that for any open disk $D$ with center $0$ and radius $\epsilon >0$, there exists an open disk $D^*\subset A$ with center $0$ and radius $0< \delta < \epsilon$. Now, use the fact that $D^*$ is open, contains $0$, and $0\in \text{acc}(B)$.