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It’s a very important space to some people, which is why we (I) write about it (at its own page). But is it important as a Baire space? (For example, if every quotient space of a Baire space were Baire, then this would be important, and it would follow that every Polish space is Baire.)
In other words, the Baire category theorem for complete metric spaces is important, and the Baire category theorem for locally compact Hausdorff spaces is important, but is the Baire category theorem for $J$ important? I don’t think so, but maybe I’m wrong about this.
Another issue is that this space is an example of a complete metric space, so why bring it up specifically in the list of examples? Only because of the coincidence of the names. (And it is a coincidence, as far as I can tell, although I would be delighted to learn otherwise.)
I still don’t understand the need for the parenthetical comment, because it’s no less important than any other example (and it’s plenty important in other contexts). It just seems like the comment might lead to some confusion in people’s minds. I think it’s enough just to disambiguate the terminology.
Maybe I’m making too big a deal about this, but I went ahead and reworded it to reflect what I think what was meant. (But I’m not as fussed about it as I was earlier.)
OK. I promoted it again to an actual example, which I assume you’re happy with.
In order to un-gray links, I gave Baire category theorem an entry, but at the moment it does nothing but point to Wikipedia.
The lead-in sentence of the Idea-section at Baire space is really not helpful, it should be changed. Instead I added pointer to “Baire category theorem” in the Examples-section, where the theorem was stated without naming it.
“A dense $G_\delta$ set (i.e. a countable intersection of dense opens) in a Baire space is a Baire space under the subspace topology. See Dan Ma’s blog, specifically Theorem 3 here.”
Dan Ma’s blog only proves that dense $G^\delta$ sets in Baire spaces are Baire, not that all $G^\delta$ sets in Baire spaces are Baire. Thus either “dense” should be added to the page or it should cite something else. Do we have a reason to believe that all $G^\delta$ sets in Baire space are Baire?
I quite fail to see what is thought to be wrong here, since the sentence as quoted does say “dense $G_\delta$ set”. Assuming the quotation is correct, I don’t see that “dense” has to be added again. (Also, please note that the convention has $\delta$ as a subscript, not as a superscript.)
I’ll check the entry to see if anything is wrong. I don’t immediately have a reference to hand that says anything about general $G_\delta$ sets in a general Baire space, although I do have one about general $G_\delta$ sets in a locally compact Hausdorff space. I had a vague memory that it held true more generally, but that would need corroboration.
You’re right, I don’t know how I missed that. Maybe it’s because I saw that above (in the second bullet point of “Examples”) it talks about a $G_\delta$ set in a locally compact Hausdorff space being Baire. Incidentally, what is your reference for that?
It’s in Munkres’s textbook, in the part about Baire spaces. Perhaps in exercises.
I’m still searching for an example those shows that the density assumption is important in the general case.
I do think the phrasing “dense $G_\delta$ set (i.e. a countable intersection of dense opens)” was a bit confusing, since “i.e.” means “in other words”, but here it applies only to the immediately preceding words “$G_\delta$ set” rather than the entire phrase “dense $G_\delta$ set”. So I changed it to “dense $G_\delta$ set (i.e. a countable intersection of dense opens that is itself dense)”.
I don’t see what’s wrong with it. In a Baire space $X$, a subset $A$ is a dense $G_\delta$ set iff it is a countable intersection of dense opens. The only possible quibble – I think a very minor one – would be the placement in the sentence of the assumption that the ambient space is Baire. In fact, I don’t like how the sentence currently reads, and so I’ll just make that adjustment.
Ah, my bad, I missed that the opens were also dense. Sorry for the noise!
With regard to a $G_\delta$ in a Baire space which is not itself a Baire space: apparently one “famous” example of a closed subspace $A$ of a Baire space $X$ that is not itself Baire is where $X$ is the space $\mathbb{R}^2 \setminus ((\mathbb{R} \setminus \mathbb{Q}) \times \{0\})$, and $A$ is the closed subspace $\mathbb{Q} \times \{0\}$. Don’t ask me why $X$ is Baire, because I haven’t figured that out yet. But, it’s not hard to see that the same $A$ is a $G_\delta$ (and if I haven’t made a mistake, in any separable metric space, every closed set is a $G_\delta$). This answers the question at the end of #7 (“no, we have no reason to believe that, because it’s false”).
I would add this to the entry, except that as I said, I don’t yet know how to show $X$ is Baire.
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