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Locally path-connected space

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A topological space in which for any point and any neighbourhood of it there is a smaller neighbourhood such that for any two points there is a continuous mapping of the unit interval into with and . Any locally path-connected space is locally connected. Any open subset of a locally path-connected space is locally path-connected. A connected locally path-connected space is a path-connected space.

Locally path-connected spaces play an important role in the theory of covering spaces. Let be a covering and let be a locally path-connected space. Then a necessary and sufficient condition for a mapping to admit a lifting, that is, a mapping such that , is that

where is the fundamental group. If is a locally simply-connected (locally -connected, see below) space and , then for any subgroup of there is a covering for which .

The higher-dimensional generalization of local path-connectedness is local -connectedness (local connectedness in dimension ). A space is said to be locally -connected if for any point and any neighbourhood of it there is a smaller neighbourhood such that any mapping of an -dimensional sphere into is homotopic in to a constant mapping. A metric space is locally -connected if and only if any mapping from an arbitrary closed subset in a metric space with can be extended to a neighbourhood of in (the Kuratowski–Dugundji theorem).


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References

[a1] J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988)
How to Cite This Entry:
Locally path-connected space. S.A. Bogatyi (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Locally_path-connected_space&oldid=19266
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098