Editing 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 [[locally connected space|local connectedness]] condition that arises in the theory of [[covering space]]s. 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 connection|Galois correspondence]] between covering spaces and [[subgroup]]s of the [[fundamental group]].

Most “nice” spaces such as [[manifold]]s and [[CW complex]]es are semi-locally simply connected, and topological spaces that do not satisfy this condition are considered somewhat [[pathological (mathematics)|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 (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.

==Examples==
[[File:Hawaiian earrings.svg|thumb|The [[Hawaiian earring]] 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 (set theory)|union]] of the [[circle]]s in the [[Euclidean plane]] with centers (1/''n'', 0) and [[radius|radii]] 1/''n'', for ''n'' a [[natural number]]. Give this space the [[subspace topology]]. Then all [[neighborhood]]s of the [[Origin (mathematics)|origin]] contain [[circle]]s 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 space|locally simply connected]]. In particular, the [[cone (topology)|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 ==
{{reflist}}
*{{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.&nbsp;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}}

[[Category:Algebraic topology]]
[[Category:Homotopy theory]]
[[Category:Properties of topological spaces]]