Not signed in (Sign In)

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-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group 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 integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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).
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeNov 19th 2016

    I added a synthetic definition of open subspace due to Penon.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 19th 2016

    I just woke up, but should that sentence end with xUx \in U instead of yUy \in U?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 19th 2016

    Yes, it should, thanks.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeNov 21st 2016
    • (edited Nov 21st 2016)

    That is highly reminiscent of the topology defined by an apartness relation. This topology is always T 1T_1, although I don't see why it would need to be T 2T_2.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeNov 30th 2016

    The first sentence of open subspace defined a subset to be open if its inclusion map was an open map. This is a bit circular given that the first sentence of open map defines a map to be open if the image of each open subspace is open. (-: I rearranged the page to start with the classical notion as part of the definition of a topological space and then proceed to generalizations to convergence spaces, locales, and so on.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeDec 2nd 2016

    That first sentence wasn't supposed to be a definition (despite being labelled as such by Urs) but an all-encompassing true statement going to the heart of the matter. If anything should be rearranged, it's open map. That said, your new version of the first sentence of open subspace (also not a definition) is actually pretty good.