The Small Inductive Dimension of Subsets of Alexandroff Spaces

Vitalij A. Chatyrko, Sang-Eon Han, Yasunao Hattori

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 $\mathbb S(k,n)=\{S\subset K^n:|S|=k+1\text{ and }ind\,S=k\}$, where $K$ is the Khalimsky line.