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
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.