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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?


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It means there is a function $j:Y\rightarrow X$ so that $jf\simeq id_X$ and $fj\simeq id_Y$. – Jason DeVito Oct 13 '11 at 16:05

Is there a specific part of the definition you don't understand? From Wikipedia:

Given two spaces $X$ and $Y$, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps $f : X → Y$ and $g : Y → X$ such that $g ∘ f$ is homotopic to the identity map id $X$ and $f ∘ g$ is homotopic to id $Y$.

The article also gives a definition of homotopy between functions

Formally, a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space X with the unit interval $[0,1]$ to $Y$ such that, if $x ∈ X$ then $H(x,0) = f(x)$ and $H(x,1) = g(x)$.

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