Homotopy

In <a href="/facts/Topology/2EqW47cU">topology</a>, two <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous functions</a> from one <a href="/facts/Topological_space/v2ut7CbK">topological space</a> to another are called homotopic (from <a href="/facts/Ancient_Greek_language/LIK7NlUh">Ancient Greek</a>: ὁμός homós 'same, similar' and τόπος tópos 'place') if one can be "continuously deformed" into the other, such a deformation being called a homotopy (/həˈmɒtəpiː/ hə-MOT-ə-pee; /ˈhoʊmoʊˌtoʊpiː/ HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition of <a href="/facts/Homotopy_groups/xjogL95Y">homotopy groups</a> and <a href="/facts/Cohomotopy_groups/BrUb4TId">cohomotopy groups</a>, important <a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariants</a> in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>.
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with <a href="/facts/Compactly_generated_space/2A5umYGa">compactly generated spaces</a>, <a href="/facts/CW_complex/rN7buLMo">CW complexes</a>, or <a href="/facts/Spectrum_(homotopy_theory)/mWho3roy">spectra</a>.

Homotopy open-in-new

Homotopy