Homotopy equivalence

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $\operatorname{id}_Y$ .

Consequences

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.