# Existence of a weak Baire space which is not Baire space

A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(X,\tau )$ is strongly nowhere dense if for any nonempty open set $U\in \tau$, there exists nonempty open set $V \subset U$ such that $V\cap A=\emptyset$.

A generalized topological space $(X,\tau )$ is called a weak Baire space if there is no nonempty open set $U\in \tau$ such that can be written as a countable union of strongly nowhere dense subsets.

Is there any weak Baire space which is not Baire space?

• I assume that Baire has the usual meaning that the intersection of countably many dense open (so in $\tau$) sets is dense (where dense = intersecting every non-empty member of $\tau$)? – Henno Brandsma Oct 23 '17 at 21:37
• @HennoBrandsma Yes. That's true. It has the usual definition. – MHenry Oct 24 '17 at 17:33
• In topological spaces, nowhere dense = strongly nowhere dense, right..? – Dominic van der Zypen Oct 25 '17 at 11:27
• @DominicvanderZypen Yes exactly. – MHenry Oct 25 '17 at 16:25

‎It is easy to see that every finite generalized topological space is a weak Baire space. Now, consider $X=\{ a,b,c \}$. Clearly, $\tau =\{ \{ a,b\}‎ , \{ ‎a,c\}‎ , \{ ‎b,c\}‎ , X, ‎\emptyset \}$ is a generalized topology on $X‎$. Then ‎$X$ is weak Baire space‎, ‎but it is not Baire space‎. Because ‎$\{ a,b \} =\{ a\} \cup‎ \{ ‎b\}$ in which $\{ a\}$ and $\{ b\}$ are nowhere dense‎.