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nLab > Latest Changes: Kolmogorov topological space

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  1.  
    • CommentRowNumber1.
    • CommentAuthormaxsnew
    • CommentTimeAug 12th 2022

    A simple characterization I just came across in some synthetic domain theory.

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2023

    I just came across

    So let’s add a reference, if not the argument itself.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeMay 16th 2023

    2.7. uses word “neighther”. Is this a legitimate English word/spelling ?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2023

    fixed spelling

    diff, v9, current

    • CommentRowNumber5.
    • CommentAuthormartinescardo
    • CommentTimeMay 16th 2023
    • (edited May 16th 2023)
    Proposition 2.1 doesn't make sense. You can't form the double exponential over the Sierpinski space unless the space X is exponentiable, and most (T0) spaces are not exponentiable (e.g. the Baire space (product of countably many copies of the discrete space of natural numbers) is a T0 non-exponentiable space).
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2023

    Just for the record: The statement of Prop. 2.1 was added in revision 6.

    This is occasion to re-iterate #2: Not to add bare statements without an indication of their justification and/or a reference.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 18th 2023

    Martin, I wonder if sense can be made of Proposition 2.1 if we consider the continuation monad unit as living in a cartesian closed category containing TopTop as a full subcategory, such as the category of pseudotopological spaces? There is a kind of intuitive sense lurking there, that T 0T_0-ness means “open sets separate points”, and this is what 2.1 is groping to express.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 19th 2023
    • (edited May 23rd 2023)

    Rather than make a meal of the suggestion of my previous comment, I replaced the wrong sentence with something that is correct: that T 0T_0-ness is equivalent to having the unit component of the standard adjunction between Frame opFrame^{op} and TopTop be a monomorphism.

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorGuest
    • CommentTimeMay 23rd 2023

    added section on Kolmogorov topological spaces in dependent type theory

    diff, v13, current

  10. Same as the case for specialization order there is material duplicated from topological space in the dependent type theory section in this article.

    Also, the definition given here about dependent type theory is really about correctly defining partial orders in dependent type theory rather than Kolmogorov topological space so I moved the technical details over to partial order and simply defined

    A Kolmogorov topological space or T 0T_0-space is a topological space which is a partial order with respect to the specialization order of the topological space.

    But then the above definition isn’t really specific to dependent type theory, so I ended up getting rid of the dependent type theory section completely and moved the definition into the alternate characterizations section as a proposition:

    Raymond

    diff, v17, current

1 to 10 of 10

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