Rings and subrings of continuous functions with countable range

Abstract

Intermediate rings of the functionally countable subalgebra of C(X) (i.e., the rings Ac (X) lying between Cc ∗ (X) and Cc (X)), where X is a Hausdorff zero-dimensional space, are studied in this article. It is shown that the structure space of each Ac (X) is homeomorphic to β 0 X, the Banaschewski compactification of X. From this a main result of [A. Veisi, ec -filters and ec -ideals in the function-ally countable subalgebra C∗ (X), Appl. Gen. Topol. 20(2) (2019), 395–405] easily follows. The countable counterpart of the m-topology and U -topology on C(X), namely mc -topology and Uc -topology, respectively, are introduced and using these, new characterizations of P -spaces and pseudocompact spaces are found out. Moreover, X is realized to be an almost P -space when and only when each maximal ideal/z-ideal in Cm(X) become a z 0-ideal. This leads to a characterization of Cc (X) among its intermediate rings for the case that X is an almost P -space. Noetherianness/Artinianness of Cc (X) and a few chosen subrings of Cc (X) are examined and finally, a complete description of z 0-ideals in a typical ring Ac (X) via z 0-ideals in Cc (X) is established.