Inessentiality with respect to subspaces
Volume 147 / 1995
Abstract
Let X be a compactum and let $A={(A_i,B_i):i=1,2,...}$ 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 $F_i$ separating $A_i$ and $B_i$ the intersection $(∩ F_i) ∩ Y $ is not empty. So A is inessential on Y if there exist closed $F_i$ separating $A_i$ and $B_i$ such that $∩ F_i$ 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.
