The Small Inductive Dimension of Subsets of Alexandroff Spaces

We describe the small inductive dimension ind in the class of Alexandroff spaces by the use of some standard spaces. Then for ind we suggest decomposition, sum and product theorems in the class. The sum and product theorems there we prove even for the small transfinite inductive dimension trind. As an application of that, for each positive integers k, n such that k <= n we get a simple description in terms of even and odd numbers of the family S(k, n) = {S subset of K-n : vertical bar S vertical bar = k + 1 and ind S = k}, where K is the Khalimsky line.