On the order hereditary closure preserving sum theorem

The main purpose of this paper is to prove the following two theorems, an order hereditary closure preserving sum theorem and an hereditary theorem: (1) If a topological property P satisfies (∑′) and is closed hereditary, and if V is an order hereditary closure preserving open cover of X and each V ∈ V is elementary and possesses P, then X possesses P. (2) Let a topological property P satisfy (∑′) and (β), and be closed hereditary. Let X be a topological space which possesses P. If every open subset G of X can be written as an order hereditary closure preserving (in G) collection of elementary sets, then every subset of X possesses P.