Difference between revisions of "Path"

Revision as of 14:52, 7 June 2020

A continuous mapping of the interval into a topological space . The points and are called the initial and the final points of the path . Given , the path defined by the formula , , is called the path inverse to and is denoted by . Given and with , the path defined by the formula

is called the composite of the paths and and is denoted by . In a path-connected space with distinguished point , the set of all paths with initial point forms the path space of .

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 , the composition defined above becomes associative, and becomes a genuine inverse to . See Fundamental groupoid.

More precisely, one may define a path as being any continuous mapping , where is called the length of the path . Then and , with of length and , are composed to , taking to and in (where has length ) to . 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

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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
-p0718101.png
-$#A+1 = 41 n = 0
-$#C+1 = 41 : ~/encyclopedia/old_files/data/P071/P.0701810 Path
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
--->
-{{TEX|auto}}
+<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>
-{{TEX|done}}
-A [[Continuous mapping|continuous mapping]]  $  f $
+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" />.
-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
-$$
-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====
-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 \} $,
+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]].
-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  $  f:  [ 0, r] \rightarrow X $,
+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).
-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====
 <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>

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Difference between revisions of "Path"

Revision as of 14:52, 7 June 2020

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