The Small Inductive Dimension of Subsets of Alexandro Spaces

Abstract

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 $k \leq n$ we get a simple description in terms of even and odd numbers of the family $\mathbb S(k,n) = \{S \subset K^n : |S| = k+1 \mbox{ and  } \ind S = k \}$, where $K$ is the Khalimsky line.