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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605001.png" /> in which for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605002.png" /> and any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605003.png" /> of it there is a smaller neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605004.png" /> such that for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605005.png" /> there is a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605006.png" /> of the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605007.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605008.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l0605009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050010.png" />. 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]].
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Locally path-connected spaces play an important role in the theory of covering spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050011.png" /> be a [[Covering|covering]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050012.png" /> be a locally path-connected space. Then a necessary and sufficient condition for a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050013.png" /> to admit a lifting, that is, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050015.png" />, is that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050016.png" /></td> </tr></table>
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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]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050017.png" /> is the [[Fundamental group|fundamental group]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050018.png" /> is a locally simply-connected (locally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050019.png" />-connected, see below) space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050020.png" />, then for any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050022.png" /> there is a covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050023.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050024.png" />.
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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
  
The higher-dimensional generalization of local path-connectedness is local <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050026.png" />-connectedness (local connectedness in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050027.png" />). A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050028.png" /> is said to be locally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050029.png" />-connected if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050030.png" /> and any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050031.png" /> of it there is a smaller neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050032.png" /> such that any mapping of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050033.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050034.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050035.png" /> is homotopic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050036.png" /> to a constant mapping. A metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050037.png" /> is locally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050038.png" />-connected if and only if any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050039.png" /> from an arbitrary closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050040.png" /> in a [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050041.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050042.png" /> can be extended to a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060500/l06050044.png" /> (the Kuratowski–Dugundji theorem).
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$$
 +
f _ {\#} ( \pi _ {1} ( Y , y _ {0} ) ) \
 +
\subset   p _ {\#} ( \pi _ {1} ( \widetilde{X}  , \widetilde{x}  _ {0} ) ) ,
 +
$$
  
 +
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=47698
This article was adapted from an original article by S.A. Bogatyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article