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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeDec 30th 2017

    Created double-negation shift, with a proof that it is equivalent to double-negated excluded middle.

    • CommentRowNumber2.
    • CommentAuthorJonasFrey
    • CommentTimeDec 31st 2017
    • (edited Dec 31st 2017)

    If I’m not mistaken, this principle holds in the topos of sheaves on a space (or locale) XX iff XX contains a dense locally decidable open, i.e. a dense open set UO(X)U\in O(X) such that for all VUV\subseteq U we have UV¬VU\subseteq V\cup\neg V. An example is the Sierpinski space, a counterexample is the real line. (the negation is the closure of the complement)

    • CommentRowNumber3.
    • CommentAuthorJonasFrey
    • CommentTimeDec 31st 2017
    • (edited Dec 31st 2017)

    I suppose at least in the spatial case the local decidability is equivalent to discreteness, so the condition means that XX has a dense and discrete open subspace.

    An example satisfying T_2 is the 1-point compacitification of an infinite set.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeDec 31st 2017

    If you’d like to add this to the page with a proof, feel free!

    • CommentRowNumber5.
    • CommentAuthorJonasFrey
    • CommentTimeJan 1st 2018

    Did it, at least with proof outlines. I hope I didn’t make any mistakes!

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJan 2nd 2018

    Thanks!

    • CommentRowNumber7.
    • CommentAuthorspitters
    • CommentTimeJul 17th 2018

    DNS holds in every Kripke model with finite frame.

    diff, v5, current

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