Different proofs of theorems of Michael and Worrell
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 , is used to give another proof of the invariance of paracompactness and metacompactness under continuous closed surgections.
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