Dense Sets and Complete Metric Spaces

What is a dense subset?

A subset of the real line is called dense when, for any two points a and b with a < b, there exists at least one point c in the subset such that $$ a < c < b $$.

This immediately implies that between any two distinct elements of a dense subset there lie infinitely many others.

Every dense subset is therefore an infinite set.

Example. The sets of rational numbers ℚ and real numbers ℝ are both dense in the real line equipped with its standard topology.

Not every infinite set is dense.

Nonetheless, every dense subset must be infinite.

Example. The natural numbers ℕ form an infinite set but are not dense in ℝ. Between any two consecutive natural numbers n and n+1 there is no other natural number, so ℕ appears in ℝ as a discrete and highly non-dense subset.

What is a complete metric space?

Completeness is a property of a metric space rather than a bare set. A metric space is called complete when every Cauchy sequence of its elements converges to a limit that remains inside the space. A complete space contains all the limit points demanded by its own metric structure.

In a complete metric space no convergent behavior points toward elements outside the space; nothing is “missing” from the standpoint of limits.

Example. The real numbers ℝ, equipped with the usual metric, form a complete metric space because every Cauchy sequence of real numbers converges to a real limit. The rational numbers ℚ, although dense in ℝ, are not complete. There exist Cauchy sequences of rationals whose limits are irrational, for example successive decimal approximations of √2.

Distinguishing density from completeness highlights how infinite subsets of ℝ can differ profoundly in their structural properties.

Example. Both ℝ and ℚ are infinite and dense in the real line. However, only ℝ is complete. From the perspective of metric structure, ℝ contains every limit demanded by its Cauchy sequences, whereas ℚ does not.