What is "homotopy"

n. 1 (context topology English) A continuous deformation of one continuous function to another. 2 (context topology English) A theory associating a system of groups to each topological space. 3 (context topology English) A system of groups associated to a topological space.

In topology, two continuous functions from one topological space to another are called homotopic ( Greek ὁμός (homós) = same, similar, and τόπος (tópos) = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.