On the Order Hereditary Closure Preserving Sum Theorem

Jianhua Gong

United Arab Emirates

United Arab Emirates University

Department of Mathematical Sciences

Ivan L. Reilly

New Zealand

University of Auckland

Department of Mathematics
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Accepted: 2013-11-15

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DOI: https://doi.org/10.4995/agt.2007.1892
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Keywords:

Elementary set, Order hereditary closure preserving, sum theorem

Supporting agencies:

This research was not funded

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.

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References:

S. P. Arya and M. K. Singal, More sum theorems for topological spaces, Pacific J. Math. 59 (1975), 1-7. http://dx.doi.org/10.2140/pjm.1975.59.1

S. P. Arya and M. K. Singal, On the closure preserving sum theorem, Proc. Amer. Math. Soc. 53 (1975), 518-522. http://dx.doi.org/10.1090/S0002-9939-1975-0383335-6

C. H. Dowker, Inductive-dimension of completely normal spaces, Quart. J. Math. 59 (1975) 1-7.

G. Gao, On the closure preserving sum theorems, Acta Math. Sinica 29 (1986), 58-62.

R. E. Hodel, Sum theorems for topological spaces, Pacific J. Math. 30 (1969), 59-65. http://dx.doi.org/10.2140/pjm.1969.30.59

Y. Katuta, A theorem On paracompactness of product spaces, Proc. Japan. Acad. 43 (1967), 615-618. http://dx.doi.org/10.3792/pja/1195521519

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