7
beleg
6y

I just made some intellectual insults, thought it might be helpful

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    It doesn't really make sense?
    Something like "Your closure is the entire space" or "there is only one closed set that fits around you" would be better imho.
  • 0
    @TempestasLudi although by accident I was [procrastinating while] studying topology at the time, I had the model theoretic notion in mind. Either way you should be dense to be able to satisfy the density axiom 😆
  • 0
    @beleg But... How does density have anything to do with embedding the real number line?
  • 0
    @TempestasLudi Embedding here is something that preserves structure, so the real line (the structure of reals and their usual ordering) being embedded into some other structure M mean that it is (order-) isomorphic to some submodel of M, and therefore they are elementary equivalent. That means the truth of any formula, as well as the density axiom (for all x and y where x<y there is some z such that x<z<y) carries over from the real line to M.
    I don't know (yet!) but I guess you can translate all this to the topological definitions of embedding and density.
  • 1
    Oooh. You're talking about the density of betweenness.
    The thing I know as "dense", works as follows: A subset S of a topological space (X, t) is dense if its closure equals X, i.e., if X is the smallest closed set (in (X, t)) that contains S.
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    @TempestasLudi I'm just learning topology, so I'm not sure but I guess these notions coincide in a metric space.
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    Hm... On the open interval (0, 1) (subset of R), betweenness is dense, but (0, 1) is not dense in R.
  • 5
    @TempestasLudi It's probably a transformation. A conformal mapping from the real axis into a closed space. Circle or ellipse maybe which in this case could be someone's head.
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