[[Homotopy theory MOC]]
# Homotopy equivalence

A **homotopy equivalence** is an isomorphism in $\hTop$. #m/def/homotopy 
Topological spaces $X$ and $Y$ are homotopy equivalent iff there exist $f \in \Top(X,Y)$ and $g \in \Top(Y,X)$ so that $gf \simeq \id_{X}$ and $fg \simeq \id_{Y}$ ([[Homotopy of maps]])

## Properties

- A [[Homotopy invariant]] is shared by homotopy equivalent spaces.

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