On Maximal Ideal Space of the Functionally Countable Subring of C(F)

Abstract

Let X be a Tychonoff space and F, a filter base of dense subsets of X (i.e., it is closed under finite intersection) and let C(F) = lim_S∊F C(S), where C(S) is the ring of all real-valued continuous functions on S. It is known that C(F) = ∪{C(S) : S ∈ F}. By C_c(F) (C^*_c (F)), we mean a subring of C(F) consisting of (bounded) functions with countable range. In this paper, we study M_c (M^*_c), the maximal ideal space of C_c(F) (C^*_c (F)) with the hull-kernel topology.
Equivalent topology for each of them provided. It is shown that both M_c and M^*_c are T₄-spaces. More generally, they are homeomorphic. Particularly, we prove that the maximal ideal space of Q_c(X) (q_c(X)) and the maximal ideal space of Q^*_c (X)(q^*_c (X)) are homeomorphic, where Q_c(X) (q_c(X)) is the maximal (classical) ring of quotients of C_c(X), and Q^*_c (X) (q^*_c (X)) is the subring consisting of bounded functions.

Key words and phrases. Functionally countable subring, filter base, hull-kernel topology, zero-dimensional space.

2010 Mathematics Subject Classification. 54C30, 54C40