Insertion of a measurable function. (English) Zbl 0851.54018
Summary: 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 \(\sigma\)-rings. Likewise, similar theorems about extremally disconnected spaces are true for \(\sigma\)-rings of a certain type. This \(\sigma\)-ring approach leads to general results on the existence of functions of class \(\alpha\).
MSC:
54C50 | Topology of 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 |
54C45 | \(C\)- and \(C^*\)-embedding |
26A21 | Classification of real functions; Baire classification of sets and functions |
54C30 | Real-valued functions in general topology |
54C20 | Extension of maps |