On the Order Hereditary Closure Preserving Sum Theorem

• Gong, Jianhua ^[1] ; Reilly, Ivan L. ^[2]
  1. [1] United Arab Emirates University

     United Arab Emirates University

     Emiratos Árabes Unidos

     
  2. [2] University of Auckland

     University of Auckland

     Nueva Zelanda

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 8, Nº. 2, 2007, págs. 267-272
• Idioma: inglés
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• Resumen

  • 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.