Semi-locally simply connected

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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 x in X and every neighborhood V of x has a open neighborhood U of x such that $x\in U\subset V$ 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. Equivalent to this definition, a space X is called semi-locally simply connected if every point in X has a open neighborhood U with the property that every loop in U can be 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, 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.

References

Bourbaki 2016, p. 340.

Bourbaki, Nicolas (2016). Topologie algébrique: Chapitres 1 à 4. Springer. Ch. IV pp. 339 -480. ISBN 978-3662493601.
J.S. Calcut, J.D. McCarthy Discreteness and homogeneity of the topological fundamental group Topology Proceedings, Vol. 34,(2009), pp. 339–349
Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.

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