nLab
semi-locally simply connected space

Contents

Idea

A topological space XX is (semi-)locally simply connected if every neighborhood of a point has a subneighbourhood in which loops based at the point in the subneighborhood can be contracted in XX. It is similar to but weaker than the condition that every neighborhood of a point has a subneighborhood that is simply connected. This latter condition is called local simple-connectedness.

Definition

A topological space XX is semi-locally simply-connected if it has a basis of neighbourhoods UU such that the inclusion Π 1(U)Π 1(X)\Pi_1(U) \to \Pi_1(X) of fundamental groupoids factors through the canonical functor Π 1(U)codisc(U)\Pi_1(U) \to codisc(U) to the codiscrete groupoid whose objects are the elements of UU. The condition on UU is equivalent to the condition that the homomorphism π 1(U,x)π 1(X,x)\pi_1(U, x) \to \pi_1(X, x) of fundamental groups induced by inclusion UXU \subseteq X is trivial.

  • A semi-locally simply connected space need not be locally simply connected. For a simple counterexample, take the cone on the Hawaiian earring space.

Application

Semi-local simple connectedness is the crucial condition needed to have a good theory of covering spaces, to the effect that the topos of permutation representations of the fundamental groupoid of XX is equivalent to the category of covering spaces of XX.

In topos theory

For a topos-theoretic notion of locally nn-connected see locally n-connected (infinity,1)-topos.

Revised on May 6, 2010 19:52:50 by Todd Trimble (69.118.56.215)