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
T.R. Hamlett1
, David Rose2
and Dragan Janković3
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
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].