#### Paths

Let be a topological space, which for our purposes will also be a manifold. A path is a continuous function, . Alternatively, may be used for the domain of . Keep in mind that a path is a function, not a set of points. Each point along the path is given by for some . This makes it appear as a nice generalization to the sequence of states visited when a plan from Chapter 2 is applied. Recall that there, a countable set of stages was defined, and the states visited could be represented as , , . In the current setting is used, in which replaces the stage index. To make the connection clearer, we could use instead of to obtain for each .

Steven M LaValle 2012-04-20