a homotopy equivalence
When are two spaces 'the same' for the purposes of homotopy theory? Not when there is a homeomorphism — that is too rigid — but when each can be continuously deformed into the other and back. A solid disk and a single point are not homeomorphic (one is two-dimensional, the other a point), yet you can collapse the disk to its centre and re-inflate it, and homotopy theory declares them genuinely equivalent. Homotopy equivalence is the official notion of sameness in this softer world.
A map f: X -> Y is a homotopy equivalence if there exists g: Y -> X with g ∘ f ≃ id_X and f ∘ g ≃ id_Y; here ≃ means homotopic and id denotes the identity map. The map g is a homotopy inverse — it need not be an actual inverse, only an inverse up to deformation. When such f exists we say X and Y are homotopy equivalent, or have the same homotopy type, written X ≃ Y. This is strictly weaker than homeomorphism: homeomorphic spaces are homotopy equivalent (take g = f^{-1}), but not conversely, as the disk-and-point example shows.
Homotopy equivalence is the relation under which all the standard algebraic invariants are designed to be unchanged: a homotopy equivalence induces isomorphisms on pi_1 (at corresponding basepoints), on every higher homotopy group pi_n, and on homology and cohomology. So if you only care about those invariants, you may freely replace a space by any homotopy-equivalent one — for instance replace an awkward open set by a friendlier CW complex. The catch worth stating honestly: having all the same invariants does NOT prove two spaces are homotopy equivalent; the Whitehead theorem rescues the converse only under the extra hypothesis of CW complexes and a single map inducing the isomorphisms.
The punctured plane R^2 \ {0}, the circle S^1, and the infinite cylinder S^1 x R are all homotopy equivalent: each deformation-retracts onto a circle, so each has pi_1 = Z and is interchangeable for homotopy purposes despite being three visibly different spaces.
Three spaces of the same homotopy type. Homotopy theory sees only the shared circle inside each.
A common error is to call a map a homotopy equivalence just because it induces an isomorphism on pi_1; you need a genuine homotopy inverse, and for non-simply-connected or higher-dimensional spaces pi_1 alone is far from enough.