Small Inductive Dimension of Topological Spaces. Part II

In this paper n denotes a natural number, X denotes a set, and F1, G1 denote families of subsets of X. Let us consider X, F1. We say that F1 is finite-order if and only if: (Def. 1) There exists n such that for every G1 such that G1 ⊆ F1 and n ∈ CardG1 holds ⋂ G1 is empty. Let us consider X. Observe that there exists a family of subsets of X which is finite-order and every family of subsets of X which is finite is also finite-order. Let us considerX, F1. The functor orderF1 yielding an extended real number is defined as follows: (Def. 2)(i) For everyG1 such that orderF1+1 ∈ CardG1 andG1 ⊆ F1 holds ⋂ G1 is empty and there exists G1 such that G1 ⊆ F1 but CardG1 = orderF1+1 but ⋂ G1 is non empty or G1 is empty if F1 is finite-order, (ii) orderF1 = +∞, otherwise.