a deformation retract
Picture a thick rubber annulus (a washer) and the central circle running around its middle. You can squeeze the rubber inward, sliding every point toward that circle, until the whole washer collapses onto it — and crucially, points already on the circle never move. That circle is a deformation retract of the annulus. It is the cleanest way to certify that a complicated space and a simple subspace inside it have the same homotopy type.
Let A be a subspace of X. A deformation retract of X onto A is a homotopy H: X x [0,1] -> X with H(x, 0) = x for all x, H(x, 1) in A for all x, and H(a, 1) = a for all a in A. So the deformation starts at the identity and ends with everything pushed into A. If in addition H(a, t) = a for all a in A and all t — points of A never move at any time — it is called a strong deformation retract. The end map r(x) = H(x, 1): X -> A is a retraction (r restricted to A is the identity), and the homotopy witnesses r ∘ i ≃ id_X where i: A -> X is inclusion; since i ∘ r is literally id on A after composing the other way is identity, inclusion i becomes a homotopy equivalence.
This is the workhorse tool for computing homotopy invariants: to find pi_1 of a messy space, retract it onto a simpler one. The Möbius band deformation-retracts onto its core circle, so pi_1 = Z; a graph deformation-retracts onto a wedge of circles. Be honest about the limits, though: not every subspace is a deformation retract (a retraction may exist without a homotopy realising it, or no retraction at all — there is no retraction of the disk onto its boundary circle), and 'homotopy equivalent' is strictly more general than 'one deformation-retracts onto the other,' though for reasonable (CW) spaces the two notions are tightly linked through the mapping cylinder.
R^n \ {0} strong-deformation-retracts onto the unit sphere S^{n-1} via H(x, t) = (1 - t) x + t (x / |x|), which fixes the sphere pointwise for all t and at t = 1 sends every nonzero vector to its normalisation. Hence R^n \ {0} ≃ S^{n-1}.
Radially pushing punctured Euclidean space onto its unit sphere — a strong deformation retract.
A retraction (a map r: X -> A fixing A) is weaker than a deformation retract; the latter additionally demands a homotopy from the identity to that retraction, and many subspaces admit a retraction but no deformation retract, or neither.