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A [[Topological space|topological space]] in which any two points can be joined by a continuous image of a simple arc; that is, a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718201.png" /> for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718203.png" /> of which there is a [[Continuous mapping|continuous mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718204.png" /> of the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718205.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718207.png" />. A path-connected Hausdorff space is a Hausdorff space in which any two points can be joined by a simple arc, or (what amounts to the same thing) a Hausdorff space into which any mapping of a zero-dimensional sphere is homotopic to a constant mapping. Every path-connected space is connected (cf. [[Connected space|Connected space]]). A continuous image of a path-connected space is path-connected.
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Path-connected spaces play an important role in homotopic topology. If a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718208.png" /> is path-connected and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p0718209.png" />, then the homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182011.png" /> are isomorphic, and this isomorphism is uniquely determined up to the action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182013.png" /> is a fibration with path-connected base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182014.png" />, then any two fibres have the same [[Homotopy type|homotopy type]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182015.png" /> is a weak fibration (a [[Serre fibration|Serre fibration]]) over a path-connected base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182016.png" />, then any two fibres have the same weak homotopy type.
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The multi-dimensional generalization of path connectedness is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182018.png" />-connectedness (connectedness in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182019.png" />). A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182020.png" /> is said to be connected in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182022.png" /> if any mapping of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182023.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182024.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182026.png" />, is homotopic to a constant mapping.
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A [[Topological space|topological space]] in which any two points can be joined by a continuous image of a simple arc; that is, a space  $  X $
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for any two points  $  x _ {0} $
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and  $  x _ {1} $
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of which there is a [[Continuous mapping|continuous mapping]]  $  f :  I \rightarrow X $
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of the unit interval  $  I = [ 0 , 1 ] $
 +
such that  $  f ( 0) = x _ {0} $
 +
and  $  f ( 1) = x _ {1} $.
 +
A path-connected Hausdorff space is a Hausdorff space in which any two points can be joined by a simple arc, or (what amounts to the same thing) a Hausdorff space into which any mapping of a zero-dimensional sphere is homotopic to a constant mapping. Every path-connected space is connected (cf. [[Connected space|Connected space]]). A continuous image of a path-connected space is path-connected.
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Path-connected spaces play an important role in homotopic topology. If a space  $  X $
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is path-connected and  $  x _ {0} , x _ {1} \in X $,
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then the homotopy groups  $  \pi _ {n} ( X , x _ {0} ) $
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and  $  \pi _ {n} ( X , x _ {1} ) $
 +
are isomorphic, and this isomorphism is uniquely determined up to the action of the group  $  \pi _ {1} ( X , x _ {0} ) $.
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If  $  p : E \rightarrow B $
 +
is a fibration with path-connected base  $  B $,
 +
then any two fibres have the same [[Homotopy type|homotopy type]]. If  $  p :  E \rightarrow B $
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is a weak fibration (a [[Serre fibration|Serre fibration]]) over a path-connected base  $  B $,
 +
then any two fibres have the same weak homotopy type.
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The multi-dimensional generalization of path connectedness is  $  k $-
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connectedness (connectedness in dimension $  k $).  
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A space $  X $
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is said to be connected in dimension $  k $
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if any mapping of an $  r $-
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dimensional sphere $  S  ^ {r} $
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into $  X $,  
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where $  r \leq  k $,  
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is homotopic to a constant mapping.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
  
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====Comments====
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A connected space is not necessarily path-connected. It is not true that in an arbitrary path-connected space any two points can be joined by a simple arc: consider the two-point [[Sierpinski space]]  $  \{ 0, 1 \} $
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in which  $  \{ 0 \} $
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is open and  $  \{ 1 \} $
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is not. The mapping  $  f:  I \rightarrow \{ 0, 1 \} $
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defined by
  
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$$
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f ( x)  =  \left \{
  
====Comments====
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\begin{array}{ll}
A connected space is not necessarily path-connected. It is not true that in an arbitrary path-connected space any two points can be joined by a simple arc: consider the two-point [[Sierpinski space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182027.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182028.png" /> is open and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182029.png" /> is not. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071820/p07182030.png" /> defined by
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0 & \textrm{ if }  x < 1/2 ,  \\
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1  & \textrm{ if }  x \geq  1/2 ,  \\
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\end{array}
  
<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/p/p071/p071820/p07182031.png" /></td> </tr></table>
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\right .$$
  
 
is continuous and connects 0 and 1. A space in which any two points can be joined by a simple arc is called arcwise connected. Thus, path-connected Hausdorff spaces are arcwise connected.
 
is continuous and connects 0 and 1. A space in which any two points can be joined by a simple arc is called arcwise connected. Thus, path-connected Hausdorff spaces are arcwise connected.

Latest revision as of 14:54, 7 June 2020


A topological space in which any two points can be joined by a continuous image of a simple arc; that is, a space $ X $ for any two points $ x _ {0} $ and $ x _ {1} $ of which there is a continuous mapping $ f : I \rightarrow X $ of the unit interval $ I = [ 0 , 1 ] $ such that $ f ( 0) = x _ {0} $ and $ f ( 1) = x _ {1} $. A path-connected Hausdorff space is a Hausdorff space in which any two points can be joined by a simple arc, or (what amounts to the same thing) a Hausdorff space into which any mapping of a zero-dimensional sphere is homotopic to a constant mapping. Every path-connected space is connected (cf. Connected space). A continuous image of a path-connected space is path-connected.

Path-connected spaces play an important role in homotopic topology. If a space $ X $ is path-connected and $ x _ {0} , x _ {1} \in X $, then the homotopy groups $ \pi _ {n} ( X , x _ {0} ) $ and $ \pi _ {n} ( X , x _ {1} ) $ are isomorphic, and this isomorphism is uniquely determined up to the action of the group $ \pi _ {1} ( X , x _ {0} ) $. If $ p : E \rightarrow B $ is a fibration with path-connected base $ B $, then any two fibres have the same homotopy type. If $ p : E \rightarrow B $ is a weak fibration (a Serre fibration) over a path-connected base $ B $, then any two fibres have the same weak homotopy type.

The multi-dimensional generalization of path connectedness is $ k $- connectedness (connectedness in dimension $ k $). A space $ X $ is said to be connected in dimension $ k $ if any mapping of an $ r $- dimensional sphere $ S ^ {r} $ into $ X $, where $ r \leq k $, is homotopic to a constant mapping.

References

[1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)

Comments

A connected space is not necessarily path-connected. It is not true that in an arbitrary path-connected space any two points can be joined by a simple arc: consider the two-point Sierpinski space $ \{ 0, 1 \} $ in which $ \{ 0 \} $ is open and $ \{ 1 \} $ is not. The mapping $ f: I \rightarrow \{ 0, 1 \} $ defined by

$$ f ( x) = \left \{ \begin{array}{ll} 0 & \textrm{ if } x < 1/2 , \\ 1 & \textrm{ if } x \geq 1/2 , \\ \end{array} \right .$$

is continuous and connects 0 and 1. A space in which any two points can be joined by a simple arc is called arcwise connected. Thus, path-connected Hausdorff spaces are arcwise connected.

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[a2] B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 15ff, 130
How to Cite This Entry:
Path-connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Path-connected_space&oldid=49359
This article was adapted from an original article by S.A. Bogatyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article