semilocally 1-connected

A topological space X is semilocally simply connected (also called semilocally 1-connected) if every point x element X has a neighborhood U such that any loop L:[0, 1]->U with basepoint x is homotopic to the trivial loop. The prefix semi- refers to the fact that the homotopy which takes L to the trivial loop can leave U and travel to other parts of X. 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.

covering space | fundamental group | homotopy | point-set topology | simply connected | universal cover