Semilocally Simply Connected

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.