Locally path-connected space
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
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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).
Comments
References
[a1] | J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988) |
Locally path-connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_path-connected_space&oldid=47698