Let $ X$ be a topological space, which for our purposes will also be a manifold. A path is a continuous function, $ \tau :
[0,1] \rightarrow X$. Alternatively, $ {\mathbb{R}}$ may be used for the domain of $ \tau$. Keep in mind that a path is a function, not a set of points. Each point along the path is given by $ \tau(s)$ for some $ s
\in [0,1]$. 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 $ x_1$, $ x_2$, $ \ldots $. In the current setting $ \tau(s)$ is used, in which $ s$ replaces the stage index. To make the connection clearer, we could use $ x$ instead of $ \tau$ to obtain $ x(s)$ for each $ s
\in [0,1]$.

Steven M LaValle 2012-04-20