Definition:Homotopy/Path/Path Homotopy
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Definition
Let $X$ be a topological space.
Let $f, g: \closedint 0 1 \to X$ be paths.
Let $H : \closedint 0 1 \times \closedint 0 1 \to X$ be a continuous map such that:
- $\forall s \in \closedint 0 1 : \map H {s, 0} = \map f s $
- $\forall s \in \closedint 0 1 : \map H {s, 1} = \map g s $
and:
- $\forall t \in \closedint 0 1 : \map H {0, t} = \map f 0 = \map g 0 $
- $\forall t \in \closedint 0 1 : \map H {1, t} = \map f 1 = \map g 1 $
Then $H$ is called a path homotopy between $f$ and $g$.
Also see
Sources
- 2000: James R. Munkres: Topology (2nd ed.): $\S 51$