nLab
semi-locally simply connected space

A topological space X is semi-locally simply-connected if it has a basis of neighbourhoods U such that the inclusion Π 1(U)Π 1(X) of fundamental groupoids factors through the canonical functor Π 1(U)codisc(U) to the codiscrete groupoid whose objects are the elements of U. Equivalently, if each point x has an open neighborhood U such that the homomorphism π 1(U,x)π 1(X,x) of fundamental groups induced by inclusion UX is trivial.