A topological space $X$ is semilocally simply connected if, for every point $x \in X$ there exists a neighborhood $U$ of $x$ such that the map of fundamental groups $$ \pi_1(U,x) \longrightarrow \pi_1(X,x) $$ induced by the inclusion map $U \hookrightarrow X$ is the trivial homomorphism.

A topological space $X$ is connected, locally path connected, and semilocally simply connected if and only if it has a universal cover.