Difference between revisions of "Locally path-connected space"
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| − | + | A topological space $ X $ | |
| + | in which for any point $ x \in X $ | ||
| + | and any neighbourhood $ O _ {x} $ | ||
| + | of it there is a smaller neighbourhood $ U _ {x} \subset O _ {x} $ | ||
| + | such that for any two points $ x _ {0} , x _ {1} \in U _ {x} $ | ||
| + | there is a continuous mapping $ F : I \rightarrow O _ {x} $ | ||
| + | of the unit interval $ I = [ 0 , 1 ] $ | ||
| + | into $ O _ {x} $ | ||
| + | with $ f ( 0) = x _ {0} $ | ||
| + | and $ f ( 1) = x _ {1} $. | ||
| + | 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|path-connected space]]. | ||
| − | + | Locally path-connected spaces play an important role in the theory of covering spaces. Let $ p : ( \widetilde{X} , \widetilde{x} _ {0} ) \rightarrow ( X, x _ {0} ) $ | |
| + | be a [[Covering|covering]] and let $ Y $ | ||
| + | be a locally path-connected space. Then a necessary and sufficient condition for a mapping $ f : ( Y , y _ {0} ) \rightarrow ( X , x _ {0} ) $ | ||
| + | to admit a lifting, that is, a mapping $ g : ( Y , y _ {0} ) \rightarrow ( \widetilde{X} , \widetilde{x} _ {0} ) $ | ||
| + | such that $ f = p \circ g $, | ||
| + | is that | ||
| − | + | $$ | |
| + | f _ {\#} ( \pi _ {1} ( Y , y _ {0} ) ) \ | ||
| + | \subset p _ {\#} ( \pi _ {1} ( \widetilde{X} , \widetilde{x} _ {0} ) ) , | ||
| + | $$ | ||
| + | where $ \pi _ {1} $ | ||
| + | is the [[Fundamental group|fundamental group]]. If $ X $ | ||
| + | is a locally simply-connected (locally $ 1 $- | ||
| + | connected, see below) space and $ x _ {0} \in X $, | ||
| + | then for any subgroup $ H $ | ||
| + | of $ \pi _ {1} ( X , x _ {0} ) $ | ||
| + | there is a covering $ p : ( \widetilde{X} , \widetilde{x} _ {0} ) \rightarrow ( X , x _ {0} ) $ | ||
| + | for which $ p _ {\#} (( \widetilde{X} , \widetilde{x} _ {0} ) ) = H $. | ||
| + | The higher-dimensional generalization of local path-connectedness is local $ k $- | ||
| + | connectedness (local connectedness in dimension $ k $). | ||
| + | A space $ X $ | ||
| + | is said to be locally $ k $- | ||
| + | connected if for any point $ x \in X $ | ||
| + | and any neighbourhood $ O _ {x} $ | ||
| + | of it there is a smaller neighbourhood $ U _ {x} \subset O _ {x} $ | ||
| + | such that any mapping of an $ r $- | ||
| + | dimensional sphere $ S ^ {r} $ | ||
| + | into $ U _ {x} $ | ||
| + | is homotopic in $ O _ {x} $ | ||
| + | to a constant mapping. A metric space $ X $ | ||
| + | is locally $ k $- | ||
| + | connected if and only if any mapping $ f : A \rightarrow X $ | ||
| + | from an arbitrary closed subset $ A $ | ||
| + | in a [[Metric space|metric space]] $ Y $ | ||
| + | with $ \mathop{\rm dim} Y \leq k + 1 $ | ||
| + | can be extended to a neighbourhood of $ A $ | ||
| + | in $ Y $( | ||
| + | the Kuratowski–Dugundji theorem). | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988)</TD></TR></table> | ||
Latest revision as of 22:17, 5 June 2020
A topological space $ X $
in which for any point $ x \in X $
and any neighbourhood $ O _ {x} $
of it there is a smaller neighbourhood $ U _ {x} \subset O _ {x} $
such that for any two points $ x _ {0} , x _ {1} \in U _ {x} $
there is a continuous mapping $ F : I \rightarrow O _ {x} $
of the unit interval $ I = [ 0 , 1 ] $
into $ O _ {x} $
with $ f ( 0) = x _ {0} $
and $ f ( 1) = x _ {1} $.
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 $ p : ( \widetilde{X} , \widetilde{x} _ {0} ) \rightarrow ( X, x _ {0} ) $ be a covering and let $ Y $ be a locally path-connected space. Then a necessary and sufficient condition for a mapping $ f : ( Y , y _ {0} ) \rightarrow ( X , x _ {0} ) $ to admit a lifting, that is, a mapping $ g : ( Y , y _ {0} ) \rightarrow ( \widetilde{X} , \widetilde{x} _ {0} ) $ such that $ f = p \circ g $, is that
$$ f _ {\#} ( \pi _ {1} ( Y , y _ {0} ) ) \ \subset p _ {\#} ( \pi _ {1} ( \widetilde{X} , \widetilde{x} _ {0} ) ) , $$
where $ \pi _ {1} $ is the fundamental group. If $ X $ is a locally simply-connected (locally $ 1 $- connected, see below) space and $ x _ {0} \in X $, then for any subgroup $ H $ of $ \pi _ {1} ( X , x _ {0} ) $ there is a covering $ p : ( \widetilde{X} , \widetilde{x} _ {0} ) \rightarrow ( X , x _ {0} ) $ for which $ p _ {\#} (( \widetilde{X} , \widetilde{x} _ {0} ) ) = H $.
The higher-dimensional generalization of local path-connectedness is local $ k $- connectedness (local connectedness in dimension $ k $). A space $ X $ is said to be locally $ k $- connected if for any point $ x \in X $ and any neighbourhood $ O _ {x} $ of it there is a smaller neighbourhood $ U _ {x} \subset O _ {x} $ such that any mapping of an $ r $- dimensional sphere $ S ^ {r} $ into $ U _ {x} $ is homotopic in $ O _ {x} $ to a constant mapping. A metric space $ X $ is locally $ k $- connected if and only if any mapping $ f : A \rightarrow X $ from an arbitrary closed subset $ A $ in a metric space $ Y $ with $ \mathop{\rm dim} Y \leq k + 1 $ can be extended to a neighbourhood of $ A $ in $ Y $( the Kuratowski–Dugundji theorem).
Comments
References
| [a1] | J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988) |
How to Cite This Entry:
Locally path-connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_path-connected_space&oldid=19266
Locally path-connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_path-connected_space&oldid=19266
This article was adapted from an original article by S.A. Bogatyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article