A topological space 
is semilocally simply connected (also called semilocally 1-connected)
if every point 
has a neighborhood 
such that any loop ![L:[0,1]->U](/images/equations/SemilocallySimplyConnected/Inline4.svg)
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.
