Consider constructing an infinite sample sequence over . What would be some desirable properties for this sequence? It would be nice if the sequence eventually reached every point in , but this is impossible because is uncountably infinite. Strangely, it is still possible for a sequence to get arbitrarily close to every element of (assuming ). In topology, this is the notion of denseness. Let and be any subsets of a topological space. The set is said to be dense in if (recall the closure of a set from Section 4.1.1). This means adding the boundary points to produces . A simple example is that is dense in . A more interesting example is that the set of rational numbers is both countable and dense in . Think about why. For any real number, such as , there exists a sequence of fractions that converges to it. This sequence of fractions must be a subset of . A sequence (as opposed to a set) is called dense if its underlying set is dense. The bare minimum for sampling methods is that they produce a dense sequence. Stronger requirements, such as uniformity and regularity, will be explained shortly.
Steven M LaValle 2012-04-20