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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.

Formally, a homotopy between two continuous functions *f* and *g* from a
topological space *X* to a topological space *Y* is defined to be a continuous function *H* : *X* × [0,1] → *Y* from the product of the space *X* with the unit interval [0,1] to *Y* such that, if *x* ∈ *X* then *H*(*x*,0) = *f*(*x*) and *H*(*x*,1) = *g*(*x*).

If we think of the second parameter of *H* as time then *H* describes a *continuous deformation* of *f* into *g*: at time 0 we have the function *f* and at time 1 we have the function *g*. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from *f* to *g* as the slider moves from 0 to 1, and vice versa.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Homotopy

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