OnP3-Paracompact Spaces

Khalid Al-Zoubi,Samer Al-Ghour

doi:10.1155/2007/80697

Abstract

Mashhour et al. [1] introduced the notions ofP1-paracompactness andP2-paracompactness of topological spaces in terms of preopen sets. In this paper, we introduce and investigate a weaker form of paracompactness which is calledP3-paracompact. We obtain various characterizations, properties, examples, and counterexamples concerning it and its relationships with other types of spaces. In particular, we show that if a space(X,T)is quasi-submaximal, then(X,T)is paracompact if it isP3-paracompact.

Highlights

Mashhour et al [1] used preopen sets to define P1-paracompact and P2-paracompact spaces
We introduce and investigate a weaker form of paracompactness which is called P3-paracompact
In [2], Ganster and Reilly studied more properties of such spaces and they proved that these two notions coincide for the class of T1-spaces

Summary

Introduction

Mashhour et al [1] used preopen sets to define P1-paracompact and P2-paracompact spaces. A collection ᏼ = {Pα : α ∈ I} of subsets of a space (X, T) is called p-locally finite if for each x ∈ X, there exists a preopen set Wx in (X, T) containing x and Wx intersects at most finitely many members of ᏼ. It follows from the definition and Lemma 1.1, if (X,T) is locally indiscrete, the collection ᏼ = {{x} : x ∈ X} is p-locally finite. Every p-locally finite collection ᏼ = {Pα : α ∈ I} of preopen subsets of a quasi-submaximal space (X,T) is locally finite. V intersects at most finitely many members of f (ᏼ) and f (ᏼ) is p-locally finite in (Y ,M)

P3-paracompact spaces

Operations