Metacompactness, paracompactness, and interior-preserving open covers

Abstract

In this paper metacompactness and paracompactness are characterized in terms of the existence of closure-preserving closed refinements and interior-preserving open star-refinements of interior-preserving directed open covers of a topological space. Several earlier results on metacompact spaces and paracompact spaces are obtained as corollaries to these characterizations. For a Tychonoff-space <italic>X</italic>, metacompactness of <italic>X</italic> is characterized in terms of orthocompactness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X times beta upper X"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo>×<!-- × --></mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>β<!-- β --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X\, \times \,\beta X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.