Inessentiality with respect to subspaces

Let X be a compactum and let A = {(Ai, Bi) : i = 1, 2, . . .} be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed Fi separating Ai and Bi the intersection ( ⋂ Fi) ∩ Y is not empty. So A is inessential on Y if there exist closed Fi separating Ai and Bi such that ⋂ Fi does not intersect Y . Properties of inessentiality are studied and applied to prove: Theorem. For every countable family A of pairs of disjoint open subsets of a compactum X there exists an open set G ⊂ X on which A is inessential and for every positivedimensional Y ⊂ X \G there exists an infinite subfamily B ⊂ A which is essential on Y . This theorem and its generalization provide a new approach for constructing hereditarily infinite-dimensional compacta not containing subspaces of positive dimension which are weakly infinite-dimensional or C-spaces.