dense subset

A dense subset is one that reaches everywhere: although it may be full of gaps, you can find a member of it arbitrarily close to any point of the whole space. Nothing is far from the subset — it is sprinkled finely enough to be near everything.

Formally, a subset D of a metric space X is dense if its closure is all of X. Equivalently, every non-empty open set of X contains a point of D, or: for every point x in X and every epsilon > 0 there is a point of D within distance epsilon of x. Density is exactly the condition needed to approximate any point of the space by members of the subset.

The classic example is the rational numbers Q inside the real line R: between any two reals there is a rational, so Q is dense even though it is full of holes and has measure zero. Density underlies approximation throughout analysis — for instance, continuous functions are dense in the space of integrable functions, which is why one may prove results first for nice functions and then pass to limits.