Bases for the closed sets

Question

I don't understand what the author means when he says that $F_1\cup F_2$ is an intersection of elements of $\mathcal{F}$. I would think it means that $F_1 \cup F_2 = \bigcap_{F\in \mathcal{F}} F$, but (b) says $\bigcap_{F\in \mathcal{F}} F=\emptyset$

See my answer here on how to solve 5E, and which also gives the right interpretation for (a). — Henno Brandsma, Mar 18 '20 at 22:26

Arthur · Accepted Answer · 2020-03-18T14:44:14.560

2

There is a big difference between

An intersection of elements in $\mathscr F$

and

The intersection of all elements in $\mathscr F$

We want there to be some subset $\mathscr E\subseteq \mathscr F$ so that $F_1\cup F_2 = \bigcap_{F\in\mathscr E}F$. However, many times, and in particular if $F_1$ and $F_2$ have points in common, $\mathscr E$ will not be all of $\mathscr F$

edited Mar 18 '20 at 14:44

answered Mar 18 '20 at 14:35

Arthur

199,419

did you mean generalized intersection? ie $F_1 \cup F_2 = \bigcap_{F\in\mathscr E}F $ – wessi Mar 18 '20 at 14:42
@jacques99 Yes, I meant intersection there. Sorry. – Arthur Mar 18 '20 at 14:44
OK thank you very much – wessi Mar 18 '20 at 14:46

Bases for the closed sets

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