Journal of the Australian Mathematical Society (Series A)

Research Article

Insertion of a measurable function

Wesley Kotzéa1 and Tomasz Kubiaka2

a1 Department of Mathematics, (Pure and Applied), Rhodes University, Grahamstown 6140, South Africa

a2 Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland

Abstract

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.

(Received March 04 1991)

(Revised July 03 1991)

1991 Mathematics subject classification (Amer. Math. Soc)

  • 54C50;
  • 28A05;
  • 28A20;
  • 54C20;
  • 54C45;
  • 26A21;
  • 54C30

Keywords and phrases

  • real-valued function;
  • σ-ring;
  • measurable function;
  • class α function;
  • insertion;
  • extension;
  • separation;
  • perfect space;
  • normal;
  • extremally disconnected.