Category:Definitions/Homotopy Equivalences

This category contains definitions related to Homotopy Equivalences.
Related results can be found in Category:Homotopy Equivalences.

Let $X$ and $Y$ be topological spaces.

Let $f: X \to Y$ be a continuous mapping.

Let there exist a continuous mapping $g: Y \to X$ such that:

the composite mapping $g \circ f$ is homotopic to the identity mapping $I_X$ on $X$

the composite mapping $f \circ g$ is homotopic to the identity mapping $I_Y$ on $Y$.

Then $X$ and $Y$ are homotopy equivalent.