On the Order Hereditary Closure Preserving Sum Theorem
Submitted: 2013-11-15
|Accepted: 2013-11-15
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Keywords:
Elementary set, Order hereditary closure preserving, sum theorem
Supporting agencies:
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 (Σ') and is closed hereditary, and if V is an order hereditary closure preserving open cover of X and each V ϵ V is elementary and possesses P, then X possesses P.
(2) Let a topological property P satisfy (Σ') 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.
References:
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