1
$\begingroup$

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?

$\endgroup$
4
  • $\begingroup$ 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 $\tau$)? $\endgroup$ Oct 23, 2017 at 21:37
  • $\begingroup$ @HennoBrandsma Yes. That's true. It has the usual definition. $\endgroup$
    – MHenry
    Oct 24, 2017 at 17:33
  • $\begingroup$ In topological spaces, nowhere dense = strongly nowhere dense, right..? $\endgroup$ Oct 25, 2017 at 11:27
  • $\begingroup$ @DominicvanderZypen Yes exactly. $\endgroup$
    – MHenry
    Oct 25, 2017 at 16:25

1 Answer 1

1
$\begingroup$

‎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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.