Category:Free Homotopies
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This category contains results about Free Homotopies.
Definitions specific to this category can be found in Definitions/Free Homotopies.
Let $X$ and $Y$ be topological spaces.
Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.
Then $f$ and $g$ are (freely) homotopic if and only if there exists a continuous mapping:
- $H: X \times \closedint 0 1 \to Y$
such that, for all $x \in X$:
- $\map H {x, 0} = \map f x$
and:
- $\map H {x, 1} = \map g x$
$H$ is called a (free) homotopy between $f$ and $g$ and we write:
- $f \simeq g$
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