Dense Sets - Topology, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Let  X ⊂ R. A subset S ⊂ X is called dense in X if any real number can be arbitrarily well-approximated by elements of S example, the rational numbers Q are dense in R, since every real number has rational numbers that are arbitrarily close to it

The image to the right shows this visually. There is a set X that is the big rectangle. There is subset S that can be said to be dense in X. This is because for any point x ∈ X, in this case a random point x in the larger set X, one could draw a circle around x using a random s ∈ S as the radius and some element of that circle will be in S.

Examples of Dense Sets

The canonical example of a dense subset of R is the set of rational numbers Q:

In fact, for any irrational number x_k, the set

Sα = {a + 6α | a, b ∈ Z}

is dense in R. This is harder to prove than the above example, and requires clever use of the pigeonhole principle.

Dense Sets in General Metric Spaces

One may define dense sets of general metric spaces similarly to how dense subsets of R were defined.

For example, let C[a, b] denote the set of continuous functions f : [a, 6] → R. One may give C[a, b] the structure of a metri by defining

By the extreme value theorem, this maximum exists for any two f,g ∈ C[a,b], so the distance function is well-defined. One easily check that d satisfies the axioms of a metric space.

Let P ⊂ C[a, b] be the subset consisting of polynomial functions. The Stone-Weierstrass theorem states that P is dense in Intuitively, this means any continuous function on a closed interval is well-approximated by polynomial functions!