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homotopy of paths
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(Definition)
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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
- $F(s,0)=p(s)$ for all $s \in I$
- $F(s,1)=q(s)$ for all $s \in I$
- $F(0,t)=x_{0}$ for all $t \in I$
- $F(1,t)=x_{1}$ 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.
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"homotopy of paths" is owned by RevBobo.
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Cross-references: continuous function, terminal point, initial point, paths, topological space
There are 46 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 9960 times total.
Classification:
| AMS MSC: | 55Q05 (Algebraic topology :: Homotopy groups :: Homotopy groups, general; sets of homotopy classes) |
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Pending Errata and Addenda
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