International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 3, Pages 433-442
doi:10.1155/S0161171297000598
Paracompactness with respect to an ideal
1Department of Mathematics, East Central University, Ada 74820, Oklahoma, USA
2Southeastern College of the Assemblies of God, 1000 Longfellow Blvd., Lakeland 35801, Florida, USA
3Dept. of Mathematical Sciences, Cameron University, Lawton 73505, Oklahoma, USA
Received 26 May 1995; Revised 26 October 1995
Copyright © 1997 T. R. Hamlett et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
An ideal on a set X is a nonempty collection of subsets of X closed under the operations
of subset and finite union. Given a topological space X and an ideal ℐ of subsets of X, X is defined to be
ℐ-paracompact if every open cover of the space admits a locally finite open refinement which is a cover
for all of X except for a set in ℐ. Basic results are investigated, particularly with regard to the ℐ-
paracompactness of two associated topologies generated by sets of the form U−I where U is open and
I∈ℐ and ⋃ {U|U is open and U−A∈ℐ, for some open set A}. Preservation of ℐ-paracompactness
by functions, subsets, and products is investigated. Important special cases of ℐ-paracompact spaces are
the usual paracompact spaces and the almost paracompact spaces of Singal and Arya [“On m-paracompact
spaces”, Math. Ann., 181 (1969), 119-133].