Previously, I defined a path in terms of a continuous function from a closed interval on the reals to a set of points in a topological space.

Because the function is *continuous*, by definition, the resultant image is homeomorphic to the closed interval on the reals. Because any closed interval on the reals is itself homeomorphic to the specific closed interval [0, 1] then the image of a path can be said to be homeomorphic to the real interval [0, 1].

**UPDATE (2005-06-01)**: As Michael Hudson points out in a comment, a path will only be homeomorphic to the closed interval [0, 1] if it doesn't cross over itself. Homeomorphisms require the function to be bijective, continuous and have a continuous inverse. A path that crosses over itself doesn't meet these criteria.

**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.

The original post had **5 comments** I'm in the process of migrating over.