# Different proofs of theorems of Michael and Worrell

• Terrence A. Edwards University of the District of Columbia
• James E. Joseph Howard University
• Bhamini M. P. Nayar Morgan Sate University
Keywords: paracompact spaces; metacompact spaces; collectionwise normal spaces; continuous functions; ultrafilters

### Abstract

Different proofs of theorems of E. Michael and J. M. Worrell, that paracompactness and metacompactness are closed continuous invariants are presented here. 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 $$g:X \to Y$$ is continuous, closed and onto is used to give these proofs. Also a characterization that a space is paracompact (metacompact) if and only if every ultrafilter of type $$P$$ (type $$M$$) converges [5], is used to give another proof of the invariance of paracompactness and metacompactness under continuous closed surgections.

### Author Biographies

Terrence A. Edwards, University of the District of Columbia

Washington, DC. 20008, USA

James E. Joseph, Howard University

Retired Professor, Department of Mathematics

Howard University

Washington, DC 20059, USA

Published
2019-06-11
How to Cite
Edwards, T. A., Joseph, J. E., & Nayar, B. M. P. (2019). Different proofs of theorems of Michael and Worrell. Journal of Advanced Studies in Topology, 10(1), 58-61. https://doi.org/10.20454/jast.2019.1539
Section
Original Articles