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 comma 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 finite 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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).
  1. As written, I do not believe Theorem 4.1 is true. Certainly, the coreflection exists but it is unclear why the topology generated by the connected components of the open subsets of XX is in fact a locally connected space. It is only obvious that locally connected spaces are the fixed points of this construction. Either this case was being mistaken for the locally path-connected case or the mistake was made of assuming that connected subspaces of XX still need to be connected as subspaces of R(X)R(X). Looking at the literature (Gleason’s paper “Universally locally connected refinements”) this simple refinement is used to show that the coreflection exists. However, the simple refinement and coreflection don’t seem to be the same. Rather, the coreflection is only guaranteed to be the infimum (in the lattice of topologies) of locally connected topologies larger than the topology of XX.

    Jeremy Brazas

    diff, v7, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2020

    Thanks for the alert. I forget what I was thinking there. But it looks like you already fixed it in rev 7?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 30th 2020

    I think I was the one who wrote that originally. Urs extracted that content from connected space into a separate article (and thus breaking some links in the process, if I recall correctly). Thanks for pointing out the (to me) subtle error, and for correcting it.