Generating set of a topological algebra

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A generating set S of a topological algebra (e.g., a Banach algebra) A is a subset of A such that the smallest closed subalgebra of A containing S is A itself.

Since polynomials are dense in the set C[0,1] of continuous functions on the interval [0,1], the set {x} (and any of its supersets) consisting of the function x\mapsto x is a generating set of the Banach algebra C[0,1]. However, it is not a generating set of the algebra C[0,1] (since in the definition of a generating set of an algebra the word closed is omitted).

A generating set is sometimes called a system of generators.

A structure (e.g., a topological algebra) A is called n-generated if there exists a generating set of A consisting of at most n elements. If A is n-generated for some finite n (resp., for n=1), then A is called finitely generated (resp., singly generated).