$$C(X)$$ C ( X ) Versus its Functionally Countable Subalgebra

Ghadermazi, Mostafa; Karamzadeh, O. A. S.; Namdari, M.

doi:10.1007/s41980-018-0124-8

Cited by 8 publications

(2 citation statements)

References 14 publications

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“…The subring 𝐶 * 𝑐 (𝑋) of 𝐶 (𝑋) consists of bounded elements of 𝐶 𝑐 (𝑋), i.e., 𝐶 * 𝑐 (𝑋) = 𝐶 * (𝑋) ∩ 𝐶 𝑐 (𝑋). For more results about 𝐶 𝑐 (𝑋) and 𝐶 * 𝑐 (𝑋), the reader is referred to [5,10,11,17,[20][21][22][23]. We recall that a zero-dimensional space is a Hausdorff space with a base consisting of clopen sets.…”

Section: Introductionmentioning

confidence: 99%

On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)

Keshtkar

Mohamadian²,

Namdari³

et al. 2022

CGASA

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In this article we consider some relations between the topological properties of the spaces 𝑋 and 𝑀𝑖𝑛(𝐶 𝑐 (𝑋)) with algebraic properties of 𝐶 𝑐 (𝑋). We observe that the compactness of 𝑀𝑖𝑛(𝐶 𝑐 (𝑋)) is equivalent to the von-Neumann regularity of 𝑞 𝑐 (𝑋), the classical ring of quotients of 𝐶 𝑐 (𝑋). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶 (𝑋) is a minimal prime ideal of 𝐶 𝑐 (𝑋) and in this case 𝑀𝑖𝑛(𝐶 (𝑋)) and 𝑀𝑖𝑛(𝐶 𝑐 (𝑋)) are homeomorphic spaces. We also observe that if 𝑋 is an 𝐹 𝑐 -space, then 𝑀𝑖𝑛(𝐶 𝑐 (𝑋)) is compact if and only if 𝑋 is countably basically disconnected if and only if 𝑀𝑖𝑛(𝐶 𝑐 (𝑋)) is homeomorphic with 𝛽 0 𝑋. Finally, by introducing 𝑧 • 𝑐 -ideals, countably cozero complemented spaces, we obtain some conditions on 𝑋 for which 𝑀𝑖𝑛(𝐶 𝑐 (𝑋)) becomes compact, basically disconnected and extremally disconnected.

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Section: Introductionmentioning

confidence: 99%

On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)

Keshtkar

Mohamadian²,

Namdari³

et al. 2022

CGASA

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“…The ring C c (X) was introduced and studied in [10]. This subalgebra has more attendance recently; see, for example, [1,4,11,14,17,18]. 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]).…”

Section: Introductionmentioning

confidence: 99%

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

Elyasi¹,

Estaji²,

Sarpoushi³

2020

Arch. Math. (Brno)

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Let Lc(X) = {f ∈ C(X) : C f = X}, where C f is the union of all open subsets U ⊆ X such that |f (U)| ℵ 0. In this paper, we present a pointfree topology version of Lc(X), named R c (L). We observe that R c (L) enjoys most of the important properties shared by R(L) and Rc(L), where Rc(L) is the pointfree version of all continuous functions of C(X) with countable image. The interrelation between R(L), R c (L), and Rc(L) is examined. We show that Lc(X) ∼ = R c O(X) for any space X. Frames L for which R c (L) = R(L) are characterized.

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