Paths

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 $s \in [0,1]$.