Denseness

Consider constructing an infinite sample sequence over ${\cal C}$. What would be some desirable properties for this sequence? It would be nice if the sequence eventually reached every point in ${\cal C}$, but this is impossible because ${\cal C}$ is uncountably infinite. Strangely, it is still possible for a sequence to get arbitrarily close to every element of ${\cal C}$ (assuming ${\cal C}\subseteq {\mathbb{R}}^m$). In topology, this is the notion of denseness. Let $U$ and $V$ be any subsets of a topological space. The set $U$ is said to be dense in $V$ if $\operatorname{cl}(U) = V$ (recall the closure of a set from Section 4.1.1). This means adding the boundary points to $U$ produces $V$. A simple example is that $(0,1) \subset {\mathbb{R}}$ is dense in $[0,1] \subset {\mathbb{R}}$. A more interesting example is that the set ${\mathbb{Q}}$ of rational numbers is both countable and dense in ${\mathbb{R}}$. Think about why. For any real number, such as $\pi \in {\mathbb{R}}$, there exists a sequence of fractions that converges to it. This sequence of fractions must be a subset of ${\mathbb{Q}}$. 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.