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I am hoping someone could help me show that the following statements, which define a Baire Space, are equivalent.

Defn1: Any topological space X such that the intersection of any countable collection of open dense sets is dense is called a Baire space.

I know that this statement is equivalent to the following:

1)$(A_i)$ countable collection of closed nowhere dense sets $\rightarrow \operatorname{int}(\cup_i A_i) = \emptyset$.

2)Union of countable collection of closed sets has an interior point $\rightarrow$ one of the closed sets has an interior point.

3)Union of any countable collection of closed sets with empty interior has empty interior.

Now the issue is that I came across another definition of a Baire Space. It is as follows:

Defn2: $X$ is Baire iff every nonempty open set is nonmeager.

Are definitions 1 and 2 equivalent? If so, why?

user111970
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2 Answers2

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Yes they are equivalent, because:

If $X$ is Baire then every non-empty open set is non-meager because a countable union of closed nowhere dense sets has an empty interior.

If every non-empty open set is non-meager, then no non-empty open set is contained in a countable union of closed nowhere dense sets, and thus a countable union of closed nowhere dense sets has empty interior and $X$ is Baire.

Bach
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Jack
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HINT:

If an open set O is meager then it is contained in the union of a sequence $F_n$ of closed sets with empty interior. Then the complement $G_n$ of $F_n$ is an open dense set (why?) and the intersection of $G_n$ is not dense (because it does not intersect $O$). The converse can be done similarly.

Or one can show more quickly that Defn1(3) is equivalent to Defn2.