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 ${\Pi }_1(U)\to {\Pi }_1(X)$ 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 $U$. Equivalently, if each point $x$ has an open neighborhood $U$ such that the homomorphism ${\pi }_1(U,x)\to {\pi }_1(X,x)$ of fundamental groups induced by inclusion $U\subseteq X$ is trivial.