On the Order Hereditary Closure Preserving Sum Theorem

Authors

  • Jianhua Gong United Arab Emirates University
  • Ivan L. Reilly University of Auckland

DOI:

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

Keywords:

Elementary set, Order hereditary closure preserving, sum theorem

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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Author Biographies

Jianhua Gong, United Arab Emirates University

Department of Mathematical Sciences

Ivan L. Reilly, University of Auckland

Department of Mathematics

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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How to Cite

[1]
J. Gong and I. L. Reilly, “On the Order Hereditary Closure Preserving Sum Theorem”, Appl. Gen. Topol., vol. 8, no. 2, pp. 267–272, Oct. 2007.

Issue

Section

Regular Articles