In mathematics, Dickson's lemma is a finiteness statement applying to n-tuples of natural numbers. It is a simple fact from combinatorics, which has become attributed to the American algebraist Dickson. It was certainly known earlier, for example to Gordan in his researches on invariant theory. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ... Paul Albert Gordan (April 27, 1837 – December 21, 1912) was a German mathematician. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ...

Stating it first for clarity for N², for any pair (m,n) of natural numbers we can introduce R_m,n, the 'rectangle' of numbers (r, s) with r at least m and s at least n. This is semi-infinite in the north and east directions, in the usual plane representation. The lemma then states that any union of the R_m,n is a finite union.

The generalization to N^k is the natural one, with k-tuples in place of pairs.

The statement says something about N^k as the topological space with the product topology arising from N, where the latter has the (semi-continuity) topology in which the open sets are all sets R_m defined as all n with n at least m. The 'rectangles' are by definition a base for the topology; it says finite unions give all open sets. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

As for the proof of the lemma, it can be derived directly, but a slick way is to show that it is a special case of Hilbert's basis theorem - in fact is essentially the case of ideals generated by monomials. In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ... In mathematics, a monomial is a particular kind of polynomial, having just one term. ...