The relation of path-homotopy is preserved under composition with continuous maps

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Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces; suppose that [ilmath]g_0,g_1:I\rightarrow X[/ilmath] (where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] - the unit interval) are paths and are path-homotopic and [ilmath]f:X\rightarrow Y[/ilmath] is a continuous map, then both [ilmath]f\circ g_0[/ilmath] and [ilmath]f\circ g_1[/ilmath] are path-homotopic in [ilmath]Y[/ilmath].

Proof

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See also

• A continuous map induces a homomorphism between fundamental groups - of which this theorem is the bulk of.