Ideals of functions with compact support in the integer-valued case

Abstract

Abstract For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set $$\begin{array}{} \displaystyle C_K(X,\mathbb Z)=\{f\in C(X,\mathbb Z)\mid \text{cl}_X(X\smallsetminus \mathsf Z(f))\text{ is compact}\} \end{array}$$ is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζ X for the Banaschewski compactification of X, we also explore some properties of ideals of C(X, ℤ) associated with subspaces of ζ X analogously to how one associates, for any Tychonoff space Y, subsets of β Y with ideals of C(Y).