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

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Statement

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

Proof

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See also

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