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nLab > Latest Changes: Baire space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 24th 2011
    On the page Baire space, I see a comment that the Baire space of irrationals is "not a very important example". I think it actually is a pretty important example; for people working with Polish spaces and descriptive set theory, it is often used as an archetypal example. At least I don't understand what the purpose of this parenthetical remark would be.
    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJun 27th 2011
    • (edited Jun 27th 2011)

    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 JJ 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.)

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 27th 2011
    • (edited Jun 27th 2011)

    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.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 27th 2011

    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.)

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeJun 30th 2011

    OK. I promoted it again to an actual example, which I assume you’re happy with.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2017

    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.

    • CommentRowNumber7.
    • CommentAuthorJoshua Meyers
    • CommentTimeOct 6th 2019

    “A dense G δG_\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 δG^\delta sets in Baire spaces are Baire, not that all G δG^\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 δG^\delta sets in Baire space are Baire?

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2019

    I quite fail to see what is thought to be wrong here, since the sentence as quoted does say “dense G δG_\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 δ\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 δG_\delta sets in a general Baire space, although I do have one about general G δG_\delta sets in a locally compact Hausdorff space. I had a vague memory that it held true more generally, but that would need corroboration.

    • CommentRowNumber9.
    • CommentAuthorJoshua Meyers
    • CommentTimeOct 7th 2019

    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 δG_\delta set in a locally compact Hausdorff space being Baire. Incidentally, what is your reference for that?

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 7th 2019

    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.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeOct 7th 2019

    I do think the phrasing “dense G δG_\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 δG_\delta set” rather than the entire phrase “dense G δG_\delta set”. So I changed it to “dense G δG_\delta set (i.e. a countable intersection of dense opens that is itself dense)”.

    diff, v13, current

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 7th 2019
    • (edited Oct 7th 2019)

    I don’t see what’s wrong with it. In a Baire space XX, a subset AA is a dense G δG_\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.

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 7th 2019

    It also seemed strange to me to say the Baire category theorem, followed by two separate bulleted assertions. The classical Baire category theorem was about completely metrizable spaces, so I made the entry say that.

    diff, v14, current

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeOct 7th 2019

    Ah, my bad, I missed that the opens were also dense. Sorry for the noise!

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 8th 2019

    With regard to a G δG_\delta in a Baire space which is not itself a Baire space: apparently one “famous” example of a closed subspace AA of a Baire space XX that is not itself Baire is where XX is the space ℝ 2∖((ℝ∖ℚ)×{0})\mathbb{R}^2 \setminus ((\mathbb{R} \setminus \mathbb{Q}) \times \{0\}), and AA is the closed subspace ℚ×{0}\mathbb{Q} \times \{0\}. Don’t ask me why XX is Baire, because I haven’t figured that out yet. But, it’s not hard to see that the same AA is a G δG_\delta (and if I haven’t made a mistake, in any separable metric space, every closed set is a G δG_\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 XX is Baire.

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