This article is about topology. For chemistry, see Homotopic groups.
The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark isocontours of one possible homotopy.
In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding 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 pathological spaces. Consequently most algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
MATHEMATICS
Continuity in mathematics is the first derivative equal to zero or the Boundary condition.
The "grandfather of mathematics" is often considered to be the ancient Greek mathematician Euclid. Euclid is known for his work "Elements," a mathematical and geometric treatise that has had a profound influence on the development of mathematics. His systematic approach to geometry and his emphasis on logical reasoning laid the foundation for much of modern mathematics.
It depends on who "he" is (or was).
No, but you can major in mathematics
Mark Hovey has written: 'Model categories' -- subject(s): Model categories (Mathematics), Complexes, Homotopy theory 'Axiomatic stable homotopy theory' -- subject(s): Homotopy theory
Klaus Johannson has written: 'Homotopy equivalences of 3-manifolds with boundaries' -- subject(s): Homotopy equivalences, Manifolds (Mathematics)
Myles Tierney has written: 'Categorical constructions in stable homotopy theory' -- subject(s): Categories (Mathematics), Complexes, Homotopy theory
Homotopy to Marie was created in 1982.
Hanno Ulrich has written: 'Fixed point theory of parametrized equivariant maps' -- subject(s): Fixed point theory, Homotopy theory, Mappings (Mathematics)
J. Frank Adams has written: 'Stable homotopy theory' -- subject(s): Homotopy theory
Donald M. Davis has written: 'From Representation Theory to Homotopy Groups' 'The nature and power of mathematics' -- subject(s): Cryptography, Fractals, Geometry, Non-Euclidean, Number theory
Richard M. Hain has written: 'Iterated integrals and homotopy periods' -- subject(s): Homotopy theory, Multiple integrals
James D. Stasheff has written: 'H-spaces from a homotopy point of view' -- subject(s): H-spaces, Homotopy theory
Michael Artin has written: 'Etale homotopy' -- subject(s): Homotopy theory 'Algebraic spaces' -- subject(s): Algebraic functions, Algebraic spaces
Rosa Antolini has written: 'Cubical structures and homotopy theory'
I. M. James has written: 'General topology and homotopy theory' -- subject(s): Topology 'Remarkable mathematicians' -- subject(s): Mathematicians, Biography 'The topology of Stiefel manifolds' -- subject(s): Stiefel manifolds 'Fibrewise topology' -- subject(s): Fiberings (Mathematics), Topology