Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps f_0 :X->Y and f_1 :X->Y are homotopic if there is a continuous map F:X×[0, 1]->Y such that F(x, 0) = f_0(x) and F(x, 1) = f_1(x). Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the punctured plane R^2 - 0. The puncture can be thought of as an obstacle.

homeomorphism | homotopy | homotopy class | homotopy group | homotopy type | topological space