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While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of those regularly locally ordered spaces which are connected, locally connected or Lindelöf. We prove that local orderability is hereditary on open, connected or compact subsets. A collection of interesting examples is also offered. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the fuzzy structure space of a semiring as well as of a Γ-semiring. We study separation axioms, compactness etc. in the fuzzy structure space of a semiring. Similar study has also been accomplished in the setting of a Γ-semiring S by using the nice interplay between S and its left operator semiring L. [ABSTRACT FROM AUTHOR]

In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets. [ABSTRACT FROM AUTHOR]

We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) in which there is a compact, metrizable, non-second countable, Cantor cube. This answers in the affirmative an open question by E. Wajch (2018) [15]. Furthermore, we strengthen a result in the above paper, namely "Countable products of metrizable spaces are quasi-metrizable implies van Douwen's Choice Principle", by first observing that the above topological statement implies the weak choice principle MC (ℵ 0 , ℵ 0) , i.e. "Every countably infinite set of countably infinite sets has a multiple choice function", and then by establishing that van Douwen's choice principle is strictly weaker than MC (ℵ 0 , ℵ 0) in ZF. The above independence result also resolves an open problem in P. Howard and J.E. Rubin (1998) [7] , and strengthens a result by P. Howard, K. Keremedis, H. Rubin, and J.E. Rubin (1998) [6]. [ABSTRACT FROM AUTHOR]

In set theory without the Axiom of Choice (AC), we investigate the deductive strength of the statements " every infinite Hausdorff space has a countably infinite cellular family " and " every infinite Hausdorff space has a countably infinite relatively discrete subspace ", and of variants of the above statements for certain classes of Hausdorff spaces, their mutual relationship, as well as their relationship with various weak choice principles. Among several results, we construct a model N of ZF (i.e., of Zermelo–Fraenkel set theory without AC) in which there exists a dense-in-itself, zero-dimensional, Hausdorff topology on ω such that c (ω) = s (ω) = ω (where c (ω) and s (ω) are the cellularity and the spread of ω , respectively), but ω has no infinite discrete subsets in N , and thus neither has infinite cellular families in N. [ABSTRACT FROM AUTHOR]