Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

Samer Al Ghour

doi:10.3390/axioms10040339

Abstract

As a weaker form of ω-paracompactness, the notion of σ-ω-paracompactness is introduced. Furthermore, as a weaker form of σ-ω-paracompactness, the notion of feebly ω-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly ω-paracompact. Furthermore, it is proven hereinthat countably ω-paracompact σ-ω-paracompact topological spaces are ω-paracompact. Furthermore, it is proven hereinthat σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat ω-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if A is a locally finite closed covering of a topological space X,τ with each A∈A being ω-paracompact and normal, then X,τ is ω-paracompact and normal, and as a corollary, a sum theorem for ω-paracompact normal topological spaces follows. Moreover, three open questions are raised.

Highlights

Generalizing the properties of the bounded and closed subsets of Rn is the main motivation for introducing compactness into the topology
Compactness and metrizability are the heartbeat of general topology
We prove that σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, ω-paracompactness is inverse invariant under perfect mappings with countable fibers

Summary

Introduction

Generalizing the properties of the bounded and closed subsets of Rn is the main motivation for introducing compactness into the topology. We prove that locally countable topological spaces are feebly ω-paracompact. A topological space (X, τ) is called countably metacompact if every countable open cover of X has a point finite open refinement.

Results

Conclusion