Semi-locally simply connected

In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space "X" is semi-locally simply connected if there is a lower bound on the sizes of the “holes” in "X". This condition is necessary for most of the theory of covering spaces, including the existence of a universal cover and the Galois correspondence between covering spaces and subgroups of the fundamental group.
Most “nice” spaces such as manifolds and CW complexes are semi-locally simply connected, and topological spaces that do not satisfy this condition are considered somewhat pathological. The standard example of a non-semi-locally simply connected space is the Hawaiian earring.
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 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 (see #Examples).
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, a condition known as unloopable ("délaçable" in French). In particular, this condition is necessary for a space to have a simply connected covering space.
Examples.
A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/"n", 0) and radii 1/"n", for "n" a natural number. Give this space the subspace topology. Then all neighborhoods of the origin contain circles that are not nullhomotopic.
The Hawaiian earring can also be used to construct a semi-locally simply connected space that is not locally simply connected. In particular, the cone on the Hawaiian earring is contractible and therefore semi-locally simply connected, but it is clearly not locally simply connected.
Topology of fundamental group.
In terms of the natural topology on the fundamental group, a locally path-connected space is semi-locally simply connected if and only if its quasitopological fundamental group is discrete.