Semi-locally simply connected
{{Expert needed|1=Mathematics|date=June 2020|reason=Appears to be too technical for a non-expert}}
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 an open neighborhood U of x such that 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).{{sfn|Bourbaki|2016|p=340}} In particular, this condition is necessary for a space to have a simply connected covering space.
Examples
File:Hawaiian earrings.svg is not semi-locally simply connected.]]
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.{{Citation needed|date = March 2016}}
References
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- {{cite book|first=Nicolas|last=Bourbaki|author-link=Nicolas Bourbaki|year=2016|title=Topologie algébrique: Chapitres 1 à 4|at=Ch. IV pp. 339 -480|publisher=Springer|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
- {{cite book | first = Allen | last = Hatcher | author-link = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = http://pi.math.cornell.edu/~hatcher/AT/ATpage.html}}