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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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$#C+1 = 29 : ~/encyclopedia/old_files/data/P071/P.0701820 Path\AAhconnected space
 
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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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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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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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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 ] $
 
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such that  $  f ( 0) = x _ {0} $
 
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and  $  f ( 1) = x _ {1} $.  
 
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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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====References====
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is path-connected and  $  x _ {0} , x _ {1} \in X $,
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<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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then the homotopy groups $  \pi _ {n} ( X , x _ {0} ) $
 
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and  $  \pi _ {n} ( X , x _ {1} ) $
 
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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 $
 
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is a fibration with path-connected base  $  B $,
 
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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 $,
 
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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.
 
  
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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
  
 
====Comments====
 
====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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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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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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<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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f ( x)  = \left \{
 
  
 
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.

Revision as of 14:52, 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 for any two points and of which there is a continuous mapping of the unit interval such that and . 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 is path-connected and , then the homotopy groups and are isomorphic, and this isomorphism is uniquely determined up to the action of the group . If is a fibration with path-connected base , then any two fibres have the same homotopy type. If is a weak fibration (a Serre fibration) over a path-connected base , then any two fibres have the same weak homotopy type.

The multi-dimensional generalization of path connectedness is -connectedness (connectedness in dimension ). A space is said to be connected in dimension if any mapping of an -dimensional sphere into , where , 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 in which is open and is not. The mapping defined by

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
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