A topological space is semilocally simply connected (also called semilocally 1-connected) if every point has a neighborhood such that any loop with basepoint is homotopic to the trivial loop. The prefix semi- refers to the fact that the homotopy which takes to the trivial loop can leave and travel to other parts of .
The property of semilocal simple connectedness is important because it is a necessary and sufficient condition for a connected, locally pathwise-connected space to have a universal cover.