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  1. I changed the name of discrete space to discrete object such that it is now consistent with codiscrete object.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 29th 2012

    Okay, good point.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMar 1st 2012

    I added a hatnote.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 2nd 2016

    In discrete object, I saw two mentions of the diagonal map “X×XXX \times X \to X”, so I made them both XX×XX \to X \times X.

    The first paragraph under Discrete Geometric Spaces puzzled me, where it says, “the converse holds if XX satisfies the T 0T_0 separation axiom” (i.e., if the diagonal map is open, then XX is discrete provided we assume T 0T_0). I don’t understand why we need that assumption.

    Suppose XX×XX \to X \times X is an open map. In particular the image of the diagonal map is an open set in X×XX \times X, i.e., for each (x,x)(x, x) there is a basic open U×VU \times V containing (x,x)(x, x) that is entirely contained in the diagonal. Thus the subset {x}×V\{x\} \times V of U×VU \times V would also be entirely contained in the diagonal, i.e., (x,y){x}×V(x, y) \in \{x\} \times V implies x=yx = y, for any yVy \in V. So the open VV is the singleton {x}\{x\}. (By similar reasoning, UU is also the singleton {x}\{x\}.) So {x}\{x\} is open, for every xXx \in X. No separation axiom needed. Am I missing something?

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeMar 2nd 2016

    A topological space has open diagonal if and only if it is discrete, indeed. I prefer this argument: for every xXx \in X, the intersection of the diagonal and {x}×X\{ x \} \times X is the singleton {(x,x)}\{ (x, x) \}, hence {x}\{ x \} is open in XX.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 2nd 2016

    Oh, I see: the inverse image of the open Δ\Delta along y(x,y)y \mapsto (x, y), for any given xXx \in X.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 2nd 2016

    I went ahead and edited that point in.