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
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)
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