Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions

Terrence Anthony Edwards,Bhamini M P Nayar,James Edwards Joseph

doi:10.11648/j.ajam.20210902.11

Terrence Anthony Edwards, Bhamini M P Nayar + Show 1 more

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https://doi.org/10.11648/j.ajam.20210902.11

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Abstract

E. Michael, in 1957, proved that the pracompactness is preserved by continuous closed functions from a space onto another. Michael’s proof is an immediate consequence of his characterization of paracompact spaces as those spaces with the property that each open cover of the space has a closure preserving refinement. Normality and transfinite induction were used to produce this characterization. J. M. Worrell, in 1985, proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact, as a consequence of a characterization of metacompact spaces he established earlier the same year. C. H. Dowker and R. N. Banerjee have provided the corresponding results for countable paracompactnes and countable metacompactness. In this article we extend these results for continuous, image closed and onto multifunctions. A result due to Joseph and Kwack that all open sets in Y have the form g(V) - g(X - V); where V is open in X, if g : X → Y is continuous, closed and onto (2006), is extended to image-closed, continuous, multifunctions. Such multifunctions as well as a characterization that a space is paracompact (metacompact) if and only if every ultrafilter of type P (M) converges, proved, in 1918, by Joseph and Nayar, is used to give generalizations of the invariance of paracompactness and metacompactness under continuous closed surjections to multifunctions.

Highlights

Introduction and PreliminariesA multifunction from a set X to a nonemty set Y is a function from X to 2Y − {∅} where 2Y is the power set of Y
Worrell proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact.[24]. This was done as a consequence of a characterization of metacompact spaces established in [23]. These theorems, as well as theorems of Dowker and Banerjee are extended to multifunctions, using a different characterization of image-closed multifunction, without using normality, transfinite induction or the wellordering principle [7], [1]
One of them is based on the generalization of the characterization of a continuous, closed and onto function provided by Joseph and Kwack to multifunctions [11]

Summary

Introduction

Introduction and PreliminariesA multifunction from a set X to a nonemty set Y is a function from X to 2Y − {∅} where 2Y is the power set of Y. Michael’s proof is an immediate consequence of his characterization of paracompact spaces as those spaces with the property that each open cover of the space has a closure preserving refinement. Worrell proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact.[24]. This was done as a consequence of a characterization of metacompact spaces established in [23]. These theorems, as well as theorems of Dowker and Banerjee are extended to multifunctions, using a different characterization of image-closed multifunction, without using normality, transfinite induction or the wellordering principle [7], [1]

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