# Doubt related to Baire Category Theorem

We know by BCT that in a complete metric space the countable intersection of open dense sets is dense. Will it also be open?

No.

Take the irrational numbers $$\Bbb{I}$$ on the real line.

We have that $$\Bbb{I}=\bigcap_{n=1}^{\infty}O_n$$ where $$O_n$$ are dense open sets.

But $$\Bbb{I}$$ is not open.

• Don't you mean intersection? – Angry_Math_Person Dec 4 '19 at 18:22
• To make things explicit, we can take $O_n=\Bbb R\setminus\frac1n\Bbb Z$ – Hagen von Eitzen Dec 4 '19 at 18:22
• @Angry_Math_Person yes it was a typo...thank you. – Marios Gretsas Dec 4 '19 at 18:23
• Ok just wondering is it possible the intersection is countable dense? – Angry_Math_Person Dec 4 '19 at 18:24
• @Angry_Math_Person No, not unless there are isolated points in the whole space. If $X$ has no isolated points and is complete metric, then the intersection will be uncountable. – Henno Brandsma Dec 4 '19 at 22:59