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Insertion of a measurable function

Part of: Maps and general types of spaces defined by maps Classical measure theory Real functions

Published online by Cambridge University Press:  09 April 2009

Wesley Kotzé  and
Tomasz Kubiak
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.

Keywords

real-valued functionσ-ringmeasurable functionclass α functioninsertionextensionseparationperfect spacenormalextremally disconnected.

MSC classification

Secondary: 54C50: Special sets defined by functions 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 54C20: Extension of maps 54C45: $C$- and $C^*$-embedding 26A21: Classification of real functions; Baire classification of sets and functions 54C30: Real-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 3 , December 1994 , pp. 295 - 304
Copyright
Copyright © Australian Mathematical Society 1994

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