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