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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2017

    I needed a way to point to the topological interval [0,1][0,1] regarded as an interval object for the use in homotopy theory. Neither the entry interval nor the entry interval object seemed specific enough for this purpose, so I created topological interval.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 1st 2017

    I added Freyd’s characterization. (This is stated elsewhere, but also belongs here.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2017

    Okay, thanks. Just for completeness, to make it the topological interval, I have added mentioning of the induced order topology to the statement of the theorem.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 1st 2017

    In some sense “interval” is a red herring because the functor could just as well be defined for bipointed topological spaces and one would get [0,1][0, 1] with its standard topology. Maybe it’s a matter of taste how one puts it, but I used intervals because the article is “topological interval”.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2017

    Okay, but then we should not say

    as an object of the category IntInt of intervals (linear orders with distinct top and bottom elements)

    but

    as an object of the category IntInt of intervals (linear orders with distinct top and bottom elements) equipped with their order topology

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 1st 2017

    Not necessarily, because then we are speaking of the “category of topological intervals” or something. That’s more structural baggage than we actually need to carry around.

    I probably could have made my point better by going minimal and removing both words, “topology” and “interval”. Just define [0,1][0, 1] as the terminal FF-coalgebra in the category of sets equipped with a distinct first and second point. Then the interval structure you want can be coinductively defined. More at coalgebra of the real interval.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2017

    All I want is that the entry titled “topological interval” says explicitly how this terminal coalgebra is a topological space, either way.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 2nd 2017

    Okay, I reworded that section and hopefully it’s satisfactory now.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2017

    Okay, thanks!

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