Examples
The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets.
By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C of continuous complex-valued functions on the interval, equipped with the supremum norm.
Every metric space is dense in its completion.
Read more about this topic: Dense Set
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