Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)