Definition
A space X is called semi-locally simply connected if every point in X has a neighborhood U with the property that every loop in U can be contracted to a single point within X (i.e. every loop is nullhomotopic). Note that 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 semi-locally simply connected if every point in X has a neighborhood U for which the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X, is trivial.
Most of the main theorems about covering spaces, including the existence of a universal cover and the Galois correspondence, require a space to be path-connected, locally path-connected, and semi-locally simply connected. In particular, this condition is necessary for a space to have a simply connected covering space.
Read more about this topic: Semi-locally Simply Connected
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