It's been a while. Back to some topology—we're almost ready to state PoincarĂ©'s Conjecture.

Consider drawing a curve on the surface of a object. If we view the surface as a topological space then the curve can be thought of as a set of points in the space with the following property: there exists a continuous function from a closed interval on the reals to that set.

This is the notion of a **path**. Some topologists will refer to the *function* as the path while others will refer to the *image* (i.e. the set of points in the space) as the path. Often it doesn't matter which is meant, e.g. in the sentence "there exists a path between any two points".

Note that there are an infinite number of continuous functions that result in the same image and vary only in the choice of parameterization.

**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.