Inessentiality with respect to subspaces

Let X be a compactum and let ${\displaystyle A=\{(A_i,B_i):i=1,2,...\}}$ 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 ${\displaystyle F_i}$ separating ${\displaystyle A_i}$ and ${\displaystyle B_i}$ the intersection ${\displaystyle (\cap F_i)\cap Y}$ is not empty. So A is inessential on Y if there exist closed ${\displaystyle F_i}$ separating ${\displaystyle A_i}$ and ${\displaystyle B_i}$ such that ${\displaystyle \cap F_i}$ does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family 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 positive-dimensional 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.