Integral extensions on rings of continuous functions
Observing that a realcompact space \(X\) is determined by the algebra \(C(X)\) of all real-valued continuous functions on \(X\) and that continuous maps between such spaces are in 1-1 correspondence with morphisms between their algebras of continuous functions, one or both of the authors have investigated in a series of papers [e.g., \textit{M. A. Mulero}, Fundam. Math. 149, No.~1, 55--66 (1996; Zbl 0840.54020); the authors, Topology Appl. 137, No.~1--3, 115--124 (2004; Zbl 1045.54007)] the algebraic properties of rings of continuous function in relation to the topological properties of mappings between topological spaces. In this paper, a surjective continuous map \(\pi : X \to Y\) between Tychonoff spaces induces an injective morphism \(C(Y) \to C(X)\) by composition: \(g \mapsto g\circ\pi\), which turns \(C(X)\) into a \(C(Y)\)-algebra. Here, \(C(Y)\) considered as a subring of \(C(X)\), the extension \(C(Y)\subseteq C(X)\) is integral if every element in \(C(X)\) satisfies a monic polynomial equation over \(C(Y)\). The main result is that if \(X\) is a compact subset of \(\mathbb R^n\), the extension \(C(Y)\subseteq C(X)\) is integral if and only if \(X\) decomposes into a finite union of closed subsets such that \(\pi\) is injective on each. In the absence of compactness, if the extension \(C(Y)\subseteq C(X)\) is integral, then the supremum of the cardinalities of the fibers of \(\pi\) is finite. The Cantor ternary set \(\mathbf {C}\) and a map \(\pi : \mathbf {C} \to I = [0, 1]\) are given and shown as an example in which a continuous onto map between compact Hausdorff spaces has an upper bound for the cardinality of the fibers of \(\pi\) but \(C(I)\subseteq C(\mathbf {C})\) is not integral.
scientific article; zbMATH DE number 5625496