PlanetMath: homotopy of maps

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homotopy of maps

(Definition)

Let be topological spaces, a closed subspace of and $f,g:X \to Y$ continuous maps. A homotopy of maps is a continuous function $F:X \times [0,1] \to Y$ satisfying

for all $x \in X$
for all $x \in X$
for all $x \in A, t\in [0,1]$ .

We say that

is homotopic to

relative to

and denote this by $f \simeq g$

. If $A=\emptyset$ , this can be written $f \simeq g$ . If

is the constant map (i.e.

for all $x \in X$ ), then we say that

is nullhomotopic.

"homotopy of maps" is owned by mathcam. [ full author list (2) | owner history (1) ]

(view preamble)

See Also: homotopy of paths, homotopy equivalence, constant function, contractible

Other names:

homotopic maps

Also defines:

homotopic, nullhomotopic

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Cross-references: constant map, continuous maps, subspace, closed, topological spaces
There are 32 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 7426 times total.

Classification:

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

Pending Errata and Addenda

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