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 \(\alpha\).