<p>The purpose of this article is to study and investigate e_c-filters on X and e_c-ideals in C^*_c(X) in which they are in fact the counterparts of z_c-filters on X and z_c-ideals in C_c(X) respectively. We show that the maximal ideals of C^*_c(X) are in one-to-one correspondence with the e_c-ultrafilters on X. In addition, the sets of e_c-ultrafilters and z_c-ultrafilters are in one-to-one correspondence. It is also shown that the sets of maximal ideals of C_c(X) and C^*_c(X) have the same cardinality. As another application of the new concepts, we characterized maximal ideals of C^*_c(X). Finally, we show that whether the space X is compact, a proper ideal I of C_c(X) is an e_c-ideal if and only if it is a closed ideal in C_c(X) if and only if it is an intersection of maximal ideals of C_c(X).</p>

In this paper, closed ideals in Cc(X), the functionally countable subalgebra of C(X), with the mc-topology, is studied. We show that ifX is CUC-space, then C*c(X) with the uniform norm-topology is a Banach algebra. Closed ideals in Cc(X) as a modified countable analogue of closed ideals in C(X) with the m-topology are characterized. For a zero-dimensional space X, we show that a proper ideal in Cc(X) is closed if and only if it is an intersection of maximal ideals of Cc(X). It is also shown that every ideal in Cc(X) with the mc-topology is closed if and only if X is a P-space if and only if every ideal in C(X) with the m-topology is closed. Moreover, for a strongly zero-dimensional space X, it is proved that a properly closed ideal in C*c(X) is an intersection of maximal ideals of C*c(X) if and only if X is pseudo compact. Finally, we show that if X is a P-space and F is an ec-filter on X, then F is an ec-ultrafilter if and only if it is a zc-ultrafilter.

<p>When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ cl_XA if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e &lt; g ∈ G and d is bounded, then d′ (A, B) &lt; g if and only if A ⊆ N_d(B, g) and B ⊆ N_d(A, g), where N_d(A, g) = {x ∈ X : d(x, A) &lt; g}, d_B(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{d_A(B), d_B(A)}.</p>

For a zero-dimensional topological space X and a totally ordered field F with interval topology, Cc(X, F) denotes the ring consisting of ordered field-valued continuous functions with countable range on X. This article aims to study and investigate the rings of quotients of Cc(X,F). Qc(X,F) (resp. qc(X,F)), the maximal (resp. classical) ring of quotients of Cc(X,F) as a modified countable analogue of Q(X) (resp. q(X)), the maximal (resp. classical) ring of quotients of C(X) are characterized. It is proved that Qc(S), the maximal ring of quotients of the subring S of Cc(X,F), is a subring of Qc(X, F) if and only if every dense ideal in S has dense cozero-set in X. Also, the coincidence of rings of quotients of Cc(X, F) is investigated. We show that qc(X,F)=Cc(X, F) if and only if the set of non-units and zero-divisors in Cc(X,F) coincide if and only if X is almost CPF-space. Finally, it is shown that the fixed ring of quotients and the cofinite ring of quotients of Cc(X) coincide if and only if Hom(Mcp) = Cc(Xp) for every p ? X.