Generating set
In mathematics, a subset S of a algebraic structure G is a generating set of G (or G is "generated" by S) if the smallest subset of G that includes S and is closed under the algebraic operations on G is G itself. For example, if G is a group and itself is the smallest subgroup of G containing S, then S is a generating set of G.
If G is a topological space, a subset S of G is said to generate G topologically if the closure of the set generated by S is G.
Examples
- The additive group of integers has 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-elements subset {3, 5} is a generating set.
- In linear algebra, S is a generating set or spanning set of a vector space V if V is the linear span of S.
- Continuous functions on the interval. Polynomials are a generating set, because closure under limits forms the entire space. (we need the concept of closure under a given topology here)