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I've been given the question:

Let $ f : X \to Y $ be a continuous map, and suppose we are given (not necessarily equal) continuous maps $ g,h : Y \to X $ such that $ gf \simeq id_X $ and $ fh \simeq id_Y $. Show that $f$ is a homotopy equivalence.

What does it mean for a single function $ f: X \to Y $ to be a homotopy equivalence?

Thanks

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    It means there is a function $j:Y\rightarrow X$ so that $jf\simeq id_X$ and $fj\simeq id_Y$.2011-10-13

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