Connectedness and compactness in C(X) with the m-topology and generalized m-topology

Connectedness of some rings of~quotients of $C(X)$ with the $m$-topology

“…Since X is a P a -space, C ra (X) = C m (X) by Proposition 2.4. Now pseudocompactness of X implies that C m (X) is connected by Proposition 3.12 in [1] and this shows that C ra (X) is also connected.…”

“…By the following lemma and proposition, we observe that whenever C m (X) is totally disconnected (equivalently, if βX \ υX is dense in βX, see [1]), then P m (X, F a ) for each infinite cardinal number a, is totally disconnected. In particular, whenever C m (X) is totally disconnected, then so is q m (X).…”

“…In [1], the authors have shown that the component of 0 in C m (X) is the ideal C ψ (X) consisting of all functions in C(X) with pseudocompact support, see [11] for more details about the ideal C ψ (X). The set {f ∈ C(X) : f g is bounded for each g ∈ C(X)} is a different characterization of C ψ (X) which is given in [9].…”

“…This is equivalent to saying that C ψ (X) is the set of all functions f ∈ C(X) such that f e is bounded for each e ∈ U + (X). In imitation of [1], it seems that the component of 0 in P m (X, F) should be of the form P ψ (X, F) = {f ∈ P (X, F) : f t is bounded, ∀ t ∈ P (X, F)} = {f ∈ P (X, F) : f e is bounded, ∀ e ∈ P + (X, F)}.…”

“…The ring C(X) equipped with the m-topology is DOI 10.14712/1213-7243.015.106 denoted by C m (X) which is a Hausdorff topological ring. Next, the m-topology on C(X) is also studied in [1], [7], [12] and [14]. In [8] the authors have defined a finer topology, namely the r-topology on C(X), based on nonnegative regular elements of C(X) instead of positive units.…”