nLab
semi-locally simply connected space

Idea

A topological space $X$ 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 $X$. 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 $X$ is semi-locally simply-connected if it has a basis of neighbourhoods $U$ such that the inclusion ${\Pi }_1(U)\to {\Pi }_1(X)$ of fundamental groupoids factors through the canonical functor ${\Pi }_1(U)\to \mathrm{codisc}(U)$ to the codiscrete groupoid whose objects are the elements of $U$. The condition on $U$ is equivalent to the condition that the homomorphism ${\pi }_1(U,x)\to {\pi }_1(X,x)$ of fundamental groups induced by inclusion $U\subseteq X$ 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 $X$ is equivalent to the category of covering spaces of $X$.

In topos theory

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