Homotopy equivalence

Two topological spaces $X$ and $Y$ are homotopy equivalent if there are continuous maps $f:X\to Y$ and $g:Y\to X$ such that $g\circ f$ is homotopic to $\operatorname{id}_X$ and $f\circ g$ is homotopic to $\operatorname{id}_Y$. A homotopy is a continuous one-parameter deformation between maps.

Homotopy equivalence preserves homotopy invariants, such as homotopy groups and singular homology, but it is weaker than homeomorphism. It records the large-scale deformation type rather than the exact local topology.