Previously we defined the notion of homotopy.

Two functions that are continuous deformations of one another are homotopic even if the two functions aren't paths.

But if the two functions *are* paths, then we can further define a stricter notion called *path homotopy*.

Two paths are **path homotopic** iff they are homotopic and they have the same start point and end point throughout the deformation.

In other words, if our paths are functions *f* and *g* from the interval [0, 1] to a topological space X, then path homotopy means not only the existence of a *continuous* map F : [0, 1] x [0, 1] -> X where

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

but also that:

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

for all t in [0, 1].

**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.