Consider two paths in the same topological space, X. Let's say one is the image of the map *f* from the interval [0, 1] and the other is the image of the map *g* from the interval [0, 1].

If it's possible to continuously deform *f* to *g* the two are said to be **homotopic**.

If *x* is the parameter for a path and *t* is the parameter for the deformation then we can think of the deformation as a *continuous* map F : [0, 1] x [0, 1] -> X where

- F(x, 0) = f(x)
- F(x, 1) = g(x)

and F(x, t) for some t, 0 < t < 1 is a path somewhere along in the deformation from *f* to *g*.

F is referred to as a **homotopy** from *f* to *g*.

Homotopies, as we shall soon see, will turn out to be a key to the topological difference between a sphere and a torus and will form the basis for our description of the PoincarĂ© Conjecture itself.

**UPDATE**: next post

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The original post was in the category: poincare_project but I'm still in the process of migrating categories over.