Paracompactness with respect to an ideal

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].

Keywords

• ideal
• compact
• paracompact
• H-closed
• quasi-H-closed
• nowhere dense
• meager
• (continuous
• almost continuous
• open
• closed
• perfect) functions
• regular closed
• open cover
• refinement
• locally finite family
• τ-boundary (τ-codense) ideal
• compatible (τ-local) ideal.