Is this correct Baire's Theorem?

[julypraise]

Homework Statement

Baire's Theorem
Let $X$ be a complete metric space. Suppose $E \subseteq X$ and

$$E = \bigcup_{n \in \mathbb{N}} F_{n}$$

where $F_{n} \subseteq X$ is closed in $X$. If all $X \backslash F_{n}$ are dense then $X \backslash E$ is dense.

Homework Equations

The Attempt at a Solution

Nothing much...

I know there may be a stronger version. But at this stage, all I need to do is to check this theorem is correct.

 

[Dick]

Looks like Baire to me. You've got a countable intersection of dense open sets, right? And you are stating the result must be dense since a complete metric space is Baire. Don't you agree?

 

[micromass]

Actually, there are two Baire theorems who state exactly the same thing. One deals with complete metric spaces, the other deals with compact Hausdorff spaces.

 

[Dick]

Actually, there are two Baire theorems who state exactly the same thing. One deals with complete metric spaces, the other deals with compact Hausdorff spaces.

Sure. Being "Baire" is a property of a topological space. Complete metric spaces aren't the only example of Baire spaces. The OP indicated it was probably a special case.

 

[julypraise]

Looks like Baire to me. You've got a countable intersection of dense open sets, right? And you are stating the result must be dense since a complete metric space is Baire. Don't you agree?

Yes the statement that you statetd, i.e., the intersection of open dense subsets is also dense, is equivalent to mine.

But is the theorem correct then?

(I've learned this from lectures and the lecturer sometimes does not specify everything like a set should not be empty or etc. So I worry about this theorem too and I've used this in my assignment too.)