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

Abstract

Let Cc(X) = {f ∈ C(X) : |f (X)| ≤ ℵ0}, C F (X) = {f ∈ C(X) : |f (X)| < ∞}, and Lc(X) = {f ∈ C(X) : C f = X}, where C f is the union of all open subsets U ⊆ X such that |f (U )| ≤ ℵ0, and CF (X) be the socle of C(X) (i.e., the sum of minimal ideals of C(X)). It is shown that if X is a locally compact space, then Lc(X) = C(X) if and only if X is locally scattered. We observe that Lc(X) enjoys most of the important properties which are shared by C(X) and Cc(X). Spaces X such that Lc(X) is regular (von Neumann) are characterized. Similarly to C(X) and Cc(X), it is shown that Lc(X) is a regular ring if and only if it is ℵ0-selfinjective. We also determine spaces X such that Soc Lc(X) = CF (X) (resp.,Ri, where Ri = R for each i, and X has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces X for which CF (X) is a prime ideal in Lc(X) are characterized and consequently for these spaces, we infer that Lc(X) can not be isomorphic to any C(Y ).

Locally functionally countable subalgebra of $\mathcal{R}(L)$

“…In [15], it was proved that L c (X) is a subalgebra as well as a sublattice of C(X) containing C c (X), and this subring is called the locally functionally countable subalgebra of C(X). The properties of the subalgebra L c (X) were mentioned in [15]. Similar to the above definition, L F (X) and L 1 (X) are the locally functionally finite and constant, respectively.…”

“…For example, let the basic neighborhood of x be the set {x}, for each point x ≥ √ 2 and for the rest of the real numbers (i.e., x < √ 2), let the basic neighborhoods be the usual open intervals containing x. This is a topology τ on R and in this case, we put X = R. Clearly, X is a completely regular Hausdorff space, which is finer than the usual topology of R. Consider the function f : X → R defined by f (x) = x for x ≥ √ 2 and f (x) = √ 2, otherwise, so we have f ∈ L c (X) \ C c (X) (for more details, see [15]). Proposition 3.…”

“…For any frame L, we put P (L) := {p ∈ L : p is a particle of L}. [15,Proposition 2.11] that if X, O(X) is a completely regular and Hausdorff topology space such that I(X), the set of isolated points of X, is dense in X, then L 1 (X) = L F (X) = L c (X) = C(X). Now, we study this result in frames.…”

“…In [16], the authors introduced and studied the ring R c (L) as the pointfree topology version of C c (X) (see also [6,8,9]). By L c (X), we mean the ring of all continuous functions that C f is dense in X for f ∈ C(X), where C f = {U : U ∈ O(X) and |f (U )| ℵ 0 }; see [15]. Note that C c (X) is the largest subring of C(X) whose elements have the countable image and that the subring L c (X) of C(X) lies between C c (X) and C(X).…”