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nLab > Latest Changes: numerable open cover

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    • CommentRowNumber1.
    • CommentAuthorarsmath
    • CommentTimeMay 11th 2020

    Mention connection with normal covers.

    diff, v10, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)
    Normal covers coincide with numerable covers.

    This is a theorem (proved in stages) due to Ernest Michael, Kiiti Morita, and Arthur H. Stone, see Morita [1, Theorem 1.2]

    [1]: Kiiti Morita.
    Paracompactness and product spaces.
    Fundamenta Mathematicae 50:3 (1962), 223–236.
    doi:10.4064/fm-50-3-223-236.
    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)
    An aside: this is the first new query box I have seen in years.

    Aren't these supposed to be posted directly to the nForum now?

    Perhaps there is a leftover instruction to post query boxes somewhere on the nLab?
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2020

    Yes, best not to use query boxes, as they will more likely be ignored and, if not, there is no good way to discuss what to do in reaction.

    The HowTo still announces query boxes here, only to say as an afterthought that they shouldn’t be used. It does so by claiming that

    the wiki software does contain a mechanism for putting questions and comments on a wiki page itself:

    which is very generous towards “the wiki software”.

    I suggest to clear the entire section HowTo – Make comments and ask questions and replace it by a pointer to the nForum.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    I am not sure about clearing the entire section, but the paragraph about query boxes should definitely be deleted.
    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020

    Added alternative characterizations.

    diff, v11, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    This comment is invalid XML; displaying source. <p>Unable to make the list format properly. Giving up.</p> <p>XML Parsing Error: mismatched tag. Expected: </li>. Location: https://ncatlab.org/nlab/show/numerable+open+cover Line Number 1173, Column 403:</p> <p><a href="https://ncatlab.org/nlab/revision/diff/numerable+open+cover/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/numerable+open+cover/11">v11</a>, <a href="https://ncatlab.org/nlab/show/numerable+open+cover">current</a></p>
    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    This is bizarre.

    Revision 11 formats correctly: https://ncatlab.org/nlab/revision/numerable+open+cover/11

    The current revision is identical to Revision 11, but does not format correctly: https://ncatlab.org/nlab/show/numerable+open+cover

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 11th 2020

    Typo, and trying a minor edit to the list to see if that fixes things.

    diff, v12, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020

    This is definitely a bug in the parser, since the interface should not even allow saving pages when XML errors are present.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 12th 2020

    OK, I fixed it by making sure the \end{proposition} was in a new paragraph. But for me, at least, none of the TeX-like syntax is rendering.

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 12th 2020

    \cite was supposed to be working, I have definitely seen it to produce something other than the TeX code.

    • CommentRowNumber13.
    • CommentAuthorarsmath
    • CommentTimeMay 12th 2020

    Found citation.

    diff, v14, current

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 12th 2020

    Added a bunch of references.

    diff, v15, current

    • CommentRowNumber15.
    • CommentAuthorarsmath
    • CommentTimeMay 14th 2020

    Query now answered, so I’m removing the query box.

    diff, v16, current

    • CommentRowNumber16.
    • CommentAuthorGuest
    • CommentTimeMay 14th 2020
    Sorry, I haven't edited nLab since the heyday of query boxes, so I wasn't aware they were deprecated.

    Thanks for the update to the article. I was confused on whether normal and numerable were the same, since one literature uses one term, and one literature uses the other. FWIW the Hoshina article has most of the details, other than the shape theory result.

    arsmath
    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2021
    • (edited Mar 22nd 2021)

    Touched the wording:

    • function ↦\mapsto continuous function;

    • support is already defined to be the closure;

    • made the forall quantifier render more intelligibly

    fixed code for a bunch of references

    diff, v17, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2022
    • (edited Jun 20th 2022)

    I am fixing the broken syntax introduced in rev 11.

    diff, v20, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2022

    added pointer to:

    diff, v21, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 17th 2022
    • (edited Jul 17th 2022)

    Concerning this paragraph (written by David Roberts, I believe):

    There is also a 1944 result by Dieudonnne that numerable covers are cofinal in locally finite covers of normal spaces — need to add this! See, eg, Theorem 6.3 of Howes’ Modern analysis and topology.

    I looked into Howes’s book and he indeed claims that Dieudonné proved that locally finite open covers of normal spaces are numerable in his 1944 paracompactness paper.

    However, Dieudonné’s paper is only 12 pages long, and after looking through it several times I do not see any mentions of partitions of unity, let alone the theorem mentioned above.

    The only results about locally finite open covers that it seems to have is Theorem 6, which shows that point-finite open covers can be refined by open covers whose closure refines the original open cover.

    diff, v22, current

    • CommentRowNumber21.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 18th 2022

    Odd! Is this even true, then?

    • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 18th 2022
    • CommentRowNumber23.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 18th 2022

    Oh, good!

1 to 23 of 23

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