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A [[Continuous mapping|continuous mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718101.png" /> of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718102.png" /> into a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718103.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718105.png" /> are called the initial and the final points of the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718106.png" />. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718107.png" />, the path defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p0718109.png" />, is called the path inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181010.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181011.png" />. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181014.png" />, the path defined by the formula
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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/p071810/p07181015.png" /></td> </tr></table>
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is called the composite of the paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181017.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181018.png" />. In a [[Path-connected space|path-connected space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181019.png" /> with distinguished point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181020.png" />, the set of all paths with initial point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181021.png" /> forms the [[Path space|path space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181022.png" />.
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A [[Continuous mapping|continuous mapping]]  $  f $
 +
of the interval  $  [ 0 , 1 ] $
 +
into a [[Topological space|topological space]] $  X $.
 +
The points  $  f ( 0) $
 +
and  $  f ( 1) $
 +
are called the initial and the final points of the path  $  f $.  
 +
Given  $  f $,
 +
the path defined by the formula  $  t \rightarrow f ( 1- t ) $,
 +
$  t \in [ 0 , 1 ] $,
 +
is called the path inverse to  $  f $
 +
and is denoted by  $  f ^ { - 1 } $.
 +
Given  $  f _ {1} $
 +
and  $  f _ {2} $
 +
with $  f _ {1} ( 1) = f _ {2} ( 0) $,
 +
the path defined by the formula
  
 +
$$
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t  \rightarrow  \left \{
  
 +
is called the composite of the paths  $  f _ {1} $
 +
and  $  f _ {2} $
 +
and is denoted by  $  f _ {1} f _ {2} $.
 +
In a [[Path-connected space|path-connected space]]  $  X $
 +
with distinguished point  $  * $,
 +
the set of all paths with initial point  $  * $
 +
forms the [[Path space|path space]] of  $  X $.
  
 
====Comments====
 
====Comments====
Generally one is interested not so much in the individual paths in a space as in the homotopy classes thereof; if one factors by the equivalence relation of homotopy relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181023.png" />, the composition defined above becomes associative, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181024.png" /> becomes a genuine inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181025.png" />. See [[Fundamental groupoid|Fundamental groupoid]].
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Generally one is interested not so much in the individual paths in a space as in the homotopy classes thereof; if one factors by the equivalence relation of homotopy relative to $  \{ 0, 1 \} $,
 +
the composition defined above becomes associative, and $  f ^ { - 1 } $
 +
becomes a genuine inverse to $  f $.  
 +
See [[Fundamental groupoid|Fundamental groupoid]].
  
More precisely, one may define a path as being any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181027.png" /> is called the length of the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181028.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181030.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181031.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181033.png" />, are composed to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181034.png" />, taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181038.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181039.png" /> has length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181040.png" />) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071810/p07181041.png" />. This composition is associative (not only homotopy associative).
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More precisely, one may define a path as being any continuous mapping $  f:  [ 0, r] \rightarrow X $,
 +
where $  r \geq  0 $
 +
is called the length of the path $  f $.  
 +
Then $  f _ {1} $
 +
and $  f _ {2} $,  
 +
with $  f _ {1} $
 +
of length $  r $
 +
and $  f _ {2} ( 0) = f _ {1} ( r) $,  
 +
are composed to $  f _ {1} f _ {2} $,  
 +
taking $  t \leq  r $
 +
to $  f _ {1} ( t) $
 +
and $  t $
 +
in $  [ r, r+ s] $(
 +
where $  f _ {2} $
 +
has length $  s $)  
 +
to $  f _ {2} ( t- r) $.  
 +
This composition is associative (not only homotopy associative).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Hilton,  S. Wylie,  "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Hilton,  S. Wylie,  "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1965)</TD></TR></table>

Revision as of 08:05, 6 June 2020


A continuous mapping $ f $ of the interval $ [ 0 , 1 ] $ into a topological space $ X $. The points $ f ( 0) $ and $ f ( 1) $ are called the initial and the final points of the path $ f $. Given $ f $, the path defined by the formula $ t \rightarrow f ( 1- t ) $, $ t \in [ 0 , 1 ] $, is called the path inverse to $ f $ and is denoted by $ f ^ { - 1 } $. Given $ f _ {1} $ and $ f _ {2} $ with $ f _ {1} ( 1) = f _ {2} ( 0) $, the path defined by the formula

$$ t \rightarrow \left \{

is called the composite of the paths $ f _ {1} $ and $ f _ {2} $ and is denoted by $ f _ {1} f _ {2} $. In a path-connected space $ X $ with distinguished point $ * $, the set of all paths with initial point $ * $ forms the path space of $ X $.

Comments

Generally one is interested not so much in the individual paths in a space as in the homotopy classes thereof; if one factors by the equivalence relation of homotopy relative to $ \{ 0, 1 \} $, the composition defined above becomes associative, and $ f ^ { - 1 } $ becomes a genuine inverse to $ f $. See Fundamental groupoid.

More precisely, one may define a path as being any continuous mapping $ f: [ 0, r] \rightarrow X $, where $ r \geq 0 $ is called the length of the path $ f $. Then $ f _ {1} $ and $ f _ {2} $, with $ f _ {1} $ of length $ r $ and $ f _ {2} ( 0) = f _ {1} ( r) $, are composed to $ f _ {1} f _ {2} $, taking $ t \leq r $ to $ f _ {1} ( t) $ and $ t $ in $ [ r, r+ s] $( where $ f _ {2} $ has length $ s $) to $ f _ {2} ( t- r) $. This composition is associative (not only homotopy associative).

References

[a1] P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1965)
How to Cite This Entry:
Path. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Path&oldid=48142
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article