A topological space 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 . 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.
A topological space is semi-locally simply-connected if it has a basis of neighbourhoods such that the inclusion of fundamental groupoids factors through the canonical functor to the codiscrete groupoid whose objects are the elements of . The condition on is equivalent to the condition that the homomorphism of fundamental groups induced by inclusion is trivial.
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 is equivalent to the category of covering spaces of .