Notation with belongs

Question

For a topological space $(X,\mathcal{F})$. Consider an open neighboorhood of $x$ called $U \subset X$. That is, $U \in \mathcal{F}$.

It is right to write $x \in U \in \mathcal{F}$? (using $\in$ twice continuosly).

Thanks in advance!

Answer 1

Yes, that is fine. I've seen similar expressions frequently, and they cannot reasonably cause misunderstanding.

If anything, the "odd" part of this notation is the fact that you write $U \in \mathcal F$ -- in practice, I rarely see anyone refer to a topology explicitly, instead preferring to call $U$ an open subset of $X$. Then the expression becomes "$x \in U \subseteq X$ with $U$ open". Of course if you have explicitly named the topology $\mathcal F$ you might as well use it.