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homotopy of maps
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(Definition)
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Let $X,Y$ be topological spaces, $A$ a closed subspace of $X$ and $f,g:X \to Y$ continuous maps. A homotopy of maps is a continuous function $F:X \times [0,1] \to Y$ satisfying
- $F(x,0)=f(x)$ for all $x \in X$
- $F(x,1)=g(x)$ for all $x \in X$
- $F(x,t)=f(x)=g(x)$ for all $x \in A, t\in [0,1]$
We say that $f$ is homotopic to $g$ relative to $A$ and denote this by $f \simeq g$ $rel A$ If $A=\emptyset$ this can be written $f \simeq g$ If $g$ is the constant map (i.e. $g(x)=y$ for all $x \in X$ , then we say that $f$ is nullhomotopic.
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"homotopy of maps" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Cross-references: constant map, continuous maps, subspace, closed, topological spaces
There are 27 references to this entry.
This is version 8 of homotopy of maps, born on 2002-01-23, modified 2008-05-27.
Object id is 1584, canonical name is HomotopyOfMaps.
Accessed 9885 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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