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Insertion of a measurable function

Published online by Cambridge University Press:  09 April 2009

Wesley Kotzé
Affiliation:
Department of Mathematics, (Pure and Applied), Rhodes University, Grahamstown 6140, South Africa
Tomasz Kubiak
Affiliation:
Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
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Abstract

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Some theorems on the existence of continuous real-valued functions on a topological space (for example, insertion, extension, and separation theorems) can be proved without involving uncountable unions of open sets. In particular, it is shown that well-known characterizations of normality (for example the Katětov-Tong insertion theorem, the Tietze extension theorem, Urysohn's lemma) are characterizations of normal σ-rings. Likewise, similar theorems about extremally disconnected spaces are true for σ-rings of a certain type. This σ-ring approach leads to general results on the existence of functions of class α.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1994

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