The classical ring of quotients of $C_c(X)$

Abstract

We construct the classical ring of quotients of the algebra of continuous real-valued functions with countable range. Our construction is a slight modification of the construction given in [3]. Dowker's example shows that the two constructions can be different.

Functionally countable subalgebras and some properties of the Banaschewski compactification

“…This characterization leads us to specify the ideal which consists of all functions in C c (X) with compact (resp., pseudocompact) support. The reader is reminded that in [1] it is claimed that it can be shown that the space of maximal ideals of C c (X) with the Stone topology (i.e., the structure space of the ring C c (X)) is isomorphic to β 0 X. We settle the internal characterization of maximal ideals of C c (X) which has this result as an immediate consequence.…”

On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

“…The proofs of most of the results in this section follow mutatis mutandis from the proofs of their corresponding results in [10]. Therefore, we state them without proofs, for the record, but give pertinent references for their corresponding proofs (note, the reason that we emphasize on the recording of these facts here is because we do believe that L c (X) and C c (X), are eligible to play appropriate roles as companions of C(X), in the future studies in the context of C(X), see for example, the comment in the first two lines of the introduction in [4]. Definition 3.1.…”

“…It turns out that C c (X), although not isomorphic to any C(Y ) in general, enjoys most of the important properties of C(X). This subalgebra has recently received some attention, see [10], [23], [24], [4], and [11]. Since C c (X) is the largest subring of C(X) whose elements have countable image, this motivates us to consider a natural subring of C(X), namely L c (X), which lies between C c (X) and C(X).…”

On a question of $C_c(X)$

“…Bhattacharjee, Knox and McGovern have found that the maximal ideal space of C c (X) is homeomorphic with β 0 X, see [4]. They remarked that the proof of this fact can be modeled after [11,Theorem 5.1].…”

“…The Gelfand-Kolmogoroff theorem states that M(Y ) is homeomorphic with the Stone-Čech compactification of Y , denoted by βY . On the other hand M c (X) is homeomorphic with the Banaschewski compactification of X, see [11], [4]. Hence β 0 X is homeomorphic with βY .…”