The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from Ancient 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 (/həˈmɒtəp/,[1] hə-MO-tə-pee; /ˈhmˌtp/,[2] HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.[3]

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

  1. ^ "Homotopy Definition & Meaning". Retrieved 22 April 2022.{{cite web}}: CS1 maint: url-status (link)
  2. ^ "Homotopy Type Theory Discussed - Computerphile". YouTube. Retrieved 22 April 2022.{{cite web}}: CS1 maint: url-status (link)
  3. ^ "Homotopy | mathematics". Encyclopedia Britannica. Retrieved 2019-08-17.

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