Separation Axioms in Topological Ordered Spaces Via b-open Sets

This paper aims to define and study new separation axioms based on the b-open sets in topological ordered spaces, namely strong -ordered spaces ( ). These new separation axioms are lying between strong -ordered spaces and spaces ( ). The implications of these new separation axioms among themselves and other existing types are studied, giving several examples and counterexamples. Also, several properties of these spaces are investigated; for example, we show that the property of strong -ordered spaces ( ) is an inherited property under open subspaces.


1-Introduction
In 1965, Nachbin [1] began the study of topological ordered spaces (Top-o.sp, for short) by associating a partially ordered relation with topological space. In 1968, McCartan [2] presentedseparation axioms ( ) inTop-o.sp's and explored several properties. He also acquired strong axioms by substituting the concept of the neighborhood with that of an open neighborhood. Arya and Gupta, in 1991, [3] presented and investigated some new separation axioms in Top-o.sp's, namely semi -ordered and semi -ordered. In 2002, Leela and Balasubramanian [4] used the notion of -
In the theme of general topology, generalized open sets play an important role and are currently considered as common research topics for many topologists worldwide. Indeed, by using generalized open sets, the significant themes of general topology and real analysis are concerned with the different modified forms of continuity, separation axioms, etc... [9,10,11]. Another type of generalized open sets in topological space, presented by Andrijevic in 1996 [12], is the proposed b-open sets. This kind of sets was studied by Ekici and Caldas [13] under the name of -open sets. Also, Caldas and Jafari [14] and Park [15] used -open sets to define -( ) and -separation axioms in topological spaces, respectively.
We, in this paper, present a new separation axiom in Top-o.sp, namely strong --ordered spaces ( ) and discuss some of their properties. Specifically, we show their relationships between themselves and with strong -ordered spaces [2] and --spaces (i = 0, 1, 2) [14,15]. Also, we give some characterizations of these new separation axioms and verify that the product of any family of strong --ordered spaces is a strong --ordered space ( ). Furthermore, we investigate the property of the strong --ordered spaces ( ) of being topological property under bijective,open, and order reversing mappings, in addition to the property of the strong --ordered space of being topological property under bijective, -closed, and quotient order mappings.

2-Preliminaries
A Top-o.sp is a triple , in which is a topological space, and is a poset. Through this paper, refers to the diagonal relation on a non-empty set . Definition 2.1 [1]. Let be a poset. Let and . Then is called an increasing set ( -set, for short), if , 6-is called a decreasing set ( -set, for short), if . Definition 2.2 [2]. A Top-o.sp is called: 1-lower (resp. upper) strong -ordered ( --ordered (resp. --ordered), for short), if for each such that , there exists an -(resp. an -) open set of (resp. ), such that (resp. ) belongs to . 2-strong -(resp. -)ordered ( --(resp. --)ordered, for short), if it is an --ordered or --ordered (resp. --ordered and --ordered). 3-strong -ordered ( --ordered, for short), if for each ; , there exist open sets and of and , respectively, where is an -set, is an -set, and . Definition 2.3 [4]. is said to be 1-strong --ordered space ( --ordered sp, for short) if it is an --ordered sp or -ordered sp. 2-strong --ordered space ( --ordered sp, for short) if it is an --ordered sp and -ordered sp. Clearly, from Definition 3.2, every strong --ordered space implies a strong --ordered space, while the opposite does not hold, as we explain in the next example.  is --sp, but the converse might not hold. The next proposition gives the condition in which the converse holds. Proposition 3.23. Let be a --sp and -space. Then, it is --ordered sp.

Proof. Let
. By is --sp, we get is -closed for each Also, from is -space, then and are -closed subsets of . From Theorem 3.7( (3) ), is -ordered sp. Proposition 3.24. Let onto, -closed, and is --sp, then is --sp.

Proof. Let
. Since is onto, then there exists such that . Since is --sp, then is -closed and, by is -closed mapping, we obtain is -closed. Hence, is -sp. Theorem 3.25. Let be a -irresolute and -closed mapping from a ---ordered sp onto where is the quotient order of induced by . Then, is --ordered sp. Proof. Since is --ordered sp, then is --sp. Since is onto and -closed, then from Proposition 3. 24, is --sp. Let be a -closed set in and is a -irresolute, then isclosed in . Since is -space, then ( ) and ( ) are -closed subsets in . Since is -closed, then ( ( )) and ( ( )) are -closed sets in . Hence, is -space. Now, is -space and --sp and, by Proposition 3.23, is --ordered sp. . Then, is --ordered sp. However, is not -ordered sp because there exists and there do not exist disjoint -open sets and in containing and respectively, where is -set and is -set. Some characterizations of --ordered sp's will be given. But before that, we need to introduce the notion of increasing (resp. decreasing) -nbhds.  an --closed --nbhd of such that . Since is injective and -closed, then is an --closed --nbhd of and implies . Hence, [ ] and, consequently, is --ordered sp. The proof in the case of is similar.  Remark 4.15. From Propositions 2.3 and 3.4, we obtain the following diagram which shows the implications between the separation axioms that we have come across in this paper and the examples that there is no other implication that holds between them.