homotopy
/ HOM-oh-toh-pee /
Homotopy is the precise word for 'continuously deforming one thing into another'. Two paths, two loops, or even two whole maps are homotopic if you can morph one into the other gradually, through a smooth family of in-between stages, with no sudden jumps. Think of a rubber band on a tabletop: any way you slide and reshape it without cutting traces out a homotopy from its start shape to its end shape. Homotopy is what makes the rubber-sheet intuition of topology into rigorous mathematics — it is the formal notion of 'I can wiggle this into that'.
Concretely, two loops a and b (both starting and ending at the same base point) are homotopic if there is a continuous family of loops, one for each time t between 0 and 1, that starts as a when t = 0 and ends as b when t = 1, every intermediate one a genuine loop pinned at the base point. Picture the whole deformation as a continuous movie: frame 0 shows loop a, the final frame shows loop b, and every frame in between is a legal loop, sliding seamlessly. If such a movie exists, a and b count as 'the same loop' for topological purposes. Being homotopic is an equivalence relation, and grouping all loops into homotopy classes is precisely the raw material of the fundamental group.
Homotopy matters because it is the right notion of sameness for the questions topology asks about holes and loops — it is exactly the deformation under which a loop's essential winding cannot change. A loop that encircles a hole and a loop that does not are never homotopic, because no continuous deformation can drag the string across the missing point. The honest distinction: homotopy is much looser than homeomorphism. A solid disk and a single point are homotopy equivalent (the disk continuously shrinks to its centre) even though they are obviously not homeomorphic — so homotopy equivalence can identify spaces that homeomorphism keeps firmly apart, and it is the coarser, more forgiving tool of the two.
In the plane with the origin removed (one missing point), take loop a that circles the origin once and loop b that stays in the right half, never enclosing the origin. Loop b is homotopic to standing still — you can shrink it to a point. But a is not homotopic to b: to deform a into b you would have to drag the loop across the missing origin, which the puncture forbids. The single removed point is detected entirely by which loops are homotopic.
Around a punctured plane, a loop circling the hole is not homotopic to one that misses it.
Homotopy equivalence is coarser than homeomorphism: a solid disk is homotopy equivalent to a point but not homeomorphic to it, so 'homotopic' identifies more spaces than 'topologically the same' does.