Small Inductive Dimension of Topological Spaces

For simplicity, we adopt the following rules: T , T1, T2 denote topological spaces, A, B denote subsets of T , F denotes a subset of T A, G, G1, G2 denote families of subsets of T , U , W denote open subsets of T A, p denotes a point of T A, n denotes a natural number, and I denotes an integer. One can prove the following propositions: (1) Fr(B ∩A) ⊆ FrB ∩A. (2) T is a T4 space if and only if for all closed subsets A, B of T such that A misses B there exist open subsets U ,W of T such that A ⊆ U and B ⊆W and U misses W . Let us consider T . The sequence of ind of T yields a sequence of subsets of 2the carrier of T and is defined by the conditions (Def. 1).