Editing Semi-locally simply connected (section)

==Definition==
A space ''X'' is called '''semi-locally simply connected''' if every [[Point (geometry)|point]] ''x'' in ''X'' and every [[Neighbourhood (mathematics)|neighborhood]] ''V'' of ''x'' has an open [[Neighbourhood (mathematics)|neighborhood]] ''U'' of ''x'' such that <math>x \in U \subset V</math> with the property that every [[Loop (graph theory)|loop]] in ''U'' can be [[homotopy|contracted]] to a single point within ''X'' (i.e.&nbsp;every loop in ''U'' is [[nullhomotopic]] in ''X''). The neighborhood ''U'' need not be [[simply connected]]: though every loop in ''U'' must be contractible within ''X'', the contraction is not required to take place inside of ''U''. For this reason, a space can be semi-locally simply connected without being [[locally simply connected]].
Equivalent to this definition, a space ''X'' is called semi-locally simply connected if every [[Point (geometry)|point]] in ''X'' has a open [[Neighbourhood (mathematics)|neighborhood]] ''U'' with the property that every [[Loop (graph theory)|loop]] in ''U'' can be [[homotopy|contracted]] to a single point within ''X'' .

Another equivalent way to define this concept is the following, a space ''X'' is semi-locally simply connected if every point in ''X'' has an open neighborhood ''U'' for which the [[homomorphism]] from the [[fundamental group]] of U to the fundamental group of ''X'', [[Fundamental group#Functoriality|induced]] by the [[inclusion map]] of ''U'' into ''X'', is trivial.

Most of the main theorems about [[covering space]]s, including the existence of a universal cover and the Galois correspondence, require a space to be [[connected space|path-connected]], [[locally connected space|locally path-connected]], and semi-locally simply connected, a condition known as '''unloopable''' (''délaçable'' in French).{{sfn|Bourbaki|2016|p=340}} In particular, this condition is necessary for a space to have a simply connected covering space.