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homotopy of paths (Definition)

Let $ X$ be a topological space and $ p,q$ paths in $ X$ with the same initial point $ x_{0}$ and terminal point $ x_{1}$. If there exists a continuous function $ F: I \times I \to X$ such that

  1. $ F(s,0)=p(s)$ for all $ s \in I$
  2. $ F(s,1)=q(s)$ for all $ s \in I$
  3. $ F(0,t)=x_{0}$ for all $ t \in I$
  4. $ F(1,t)=x_{1}$ for all $ t \in I$

we call $ F$ a homotopy of paths in $ X$ and say $ p,q$ are homotopic paths in $ X$. $ F$ is also called a continuous deformation.



"homotopy of paths" is owned by RevBobo.
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See Also: homotopy of maps, homotopy with a contractible domain, (path) connectness as a homotopy invariant

Other names:  homotopic paths, continuous deformation, homotopy
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Cross-references: continuous function, terminal point, initial point, paths, topological space
There are 38 references to this entry.

This is version 4 of homotopy of paths, born on 2002-01-23, modified 2002-02-12.
Object id is 1576, canonical name is HomotopyOfPaths.
Accessed 7589 times total.

Classification:
AMS MSC: 55Q05 (Algebraic topology :: Homotopy groups :: Homotopy groups, general; sets of homotopy classes)
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