Let be a collection of open sets of real numbers. Suppose is such that .

Show that

I've managed to prove that .

How can i prove the other inclusion?

Thanks in advance.

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- Feb 8th 2010, 02:53 AMJoachimAgrell[SOLVED] Topology
Let be a collection of open sets of real numbers. Suppose is such that .

Show that

I've managed to prove that .

How can i prove the other inclusion?

Thanks in advance. - Feb 8th 2010, 07:35 AMOpalg
This looks suspiciously like the Baire category theorem (see the section headed "Proof" in that link).

- Feb 8th 2010, 07:36 PMJoachimAgrell
I actually used Baire category theorem. Here goes the solution:

Let .

Since , it follows that is a dense subset of F for all lambda. Hence each is a dense subset of F.

Suppose but . That means*x*is not an adherent point of . Therefore such that . But , so it must be true that .

On the other hand, since each is a dense subset of F, the category theorem guarantees that the countable intersection of these sets is a dense set in F. Thus , a contradiction.