Rings and subrings of continuous functions with countable range

Abstract

Intermediate rings of the functionally countable subalgebra of C(X) (i.e., the rings A_c(X) lying between C_c^∗(X) and C_c(X)), where X is a Hausdorff zero-dimensional space, are studied in this article. It is shown that the structure space of each A_c(X) is homeomorphic to β₀X, the Banaschewski compactification of X. From this a main result of [A. Veisi, e_c-filters and e_c-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 m_c-topology and U_c-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 z⁰-ideal. This leads to a characterization of C_c(X) among its intermediate rings for the case that X is an almost P -space. Noetherianness/Artinianness of C_c(X) and a few chosen subrings of C_c(X) are examined and finally, a complete description of z⁰-ideals in a typical ring A_c(X) via z⁰-ideals in C_c(X) is established.

Mathematics Subject Classification (2010):

• 54C40

Key words:

• Intermediate rings
• zero-dimensional space
• P -spaces
• almost P -spaces
• mc- topology
• z⁰-ideals
• Banaschewski compactification