|
In mathematics, specifically algebraic topology, the phrase semi-locally simply connected refers to 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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「semilocally simply connected」の詳細全文を読む スポンサード リンク
|