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nLab > Latest Changes: continuous map

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 23rd 2017
    • (edited Jan 23rd 2017)

    I gave continuous map a little bit of substance by giving it an actual Idea-paragraph and by writing out the epsilontic definition for the case of metric spaces, together with its equivalence to the “abstract” definition in terms of opens.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJan 23rd 2017

    I added the nonstandard definition, largely because it makes precise an intuitive version that can go in the Idea section.

    Something should be said about continuity at a point, but I don't have time to do that right now.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJan 24th 2017

    Another way to make precise the “preservation of closeness” is in terms of closeness to sets: if x≈A means that x belongs to the closure of A (a relation in terms of which the notion of topological space can equivalently be defined, at least in classical mathematics), then f is continuous iff x≈A implies f(x)≈f(A).

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeJan 25th 2017

    True, and of course that's directly related to the stuff you've been doing with proximity and apartness spaces.

  5. I added to continuous map Frank Waaldijk’s very nice negative result that in constructive mathematics (by which I mean the kind of mathematics possible in a topos with natural numbers object) it cannot be shown that there is a notion of continuity of set-theoretic functions such that certain natural desiderata hold.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 23rd 2017
    • (edited Feb 23rd 2017)

    That result is already listed at fan theorem (since the existence of such a notion is equivalent to the fan theorem). Which is not to say that it should not also be listed at continuous map, of course.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 23rd 2017

    I added some remarks beforehand about the different notions of continuity in constructive analysis.

    Also, the problem is not so much that a kontinuous function on [0,1] might not be bounded (since a uniformly continuous function must be bounded, even without the Fan Theorem, and it is specified that kontinuous functions on [0,1] are uniformly continuous) but that a kontinuous, positive-valued function on [0,1] might not be bounded below by a positive number (so that its reciprocal could not be bounded above, hence could not itself be kontinuous). So I rephrased that bit.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 24th 2017

    Toby, is there a significance to spelling it ’kontinuous’?

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 24th 2017

    I presume that Waaldijk used a nonstandard spelling because he was speaking about an arbitrary set (with given properties) of partial functions while wanting to invoke a sense that these were more or less the continuous functions.

  10. adding section on continuous functions in synthetic topology/cohesive homotopy type theory.

    Anonymous

    diff, v32, current

  11. adding section on continuous functions on the real numbers

    Anonymous

    diff, v32, current

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