The implications between different classes of topological spaces considered in this paper are summarized in the following diagram: cC Ace ~ AcPC cPCHere P = Para, C = Compact, c = Countably, A = Almost. The class of almost-gR-compact spaces has been introduced and studied by Singal and Singal [24].The authors are grateful to the referee for his valuable suggestions for the improvement of the paper.

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Abstract. 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. AMS Classification: 54D20