EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 3, 2020, 513-528 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global New Topology on Omega Algebra Mesfer Alqahtani1,∗, Cenap Özel1, Ibtesam Alshammari2 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, Faculty of Science, University of Hafr Al Batin, Hafr Al Batin, Saudi Arabia Abstract. The purpose of this paper is to define a new topology called omega topology over a new structure called omega algebra and discuss some of its topological properties. Four different examples of omega topology are introduced. Furthermore, we define a new topology over a semiring in conventional algebra and we study the relationship between omega topology and weaker kinds of normality. 2020 Mathematics Subject Classifications: 16Y60, 54F15 Key Words and Phrases: Tropical geometry, Idempotent semiring, Omega algebra, Omega topology, Topological space, Topological properties, Homeomorphism 1. Introduction Tropical geometry is the most recent but fast growing branch of mathematical science, which is analytically based on idempotent analysis and algebraically on idempotent semir- ings also known as tropical semirings. These are basically extended sets of real numbers R∞ : = R∪{∞} and R−∞ : = R∪{−∞} which are given monoidal structures by using min and max operations for addition, respectively. In order to adhere the semiring structure, the additive operation of R is used as the multiplication operation. By these choices, both R∞ and R−∞ become idempotent semirings. In the literature, they are also termed as min and max plus algebras, respectively. In both cases, 0 of R becomes a multiplicative identity and ∞ and −∞ become additive identities of these semirings, respectively. Inter- estingly, some authors associated R−∞ to tropical geometry, while other authors associated R∞ to tropical geometry, see for instance [8], [10], [12] and [14]. Omega algebra, or ”w- algebra” for short, unifies the different terms and introduces an original structure, which, in fact,is an ”abstract tropical algebra”. The R−∞ and R∞ and their nearby structures, like min−max and max− times algebras, etc., are all subsumed under omega algebra. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3764 Email addresses: mesfer ¯ alqhtani@hotmail.com (M. Alqahtani), cozel@kau.edu.sa (C. Özel), iealshamri@uhb.edu.sa (I. Alshammari) https://www.ejpam.com 513 c© 2020 EJPAM All rights reserved. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 514 All these are idempotent semirings, which are also called dioids. In the previous studies, for the construction of all such semirings, an ordered infinite abelian group is mandatory. In ω− algebra, the definition is extended to cyclically ordered abelian groups and also to finite sets under some suitable ordering. Note that cyclically ordered abelian groups are more general than that of ordered abelian groups [16]. The aim of this paper is to define a new topology, which is called omega topology over omega algebra, and discuss some of its topological properties. Four different examples of omega topology are introduced. Furthermore, we defined a new topology over a semiring in conventional algebra and study the relationship between omega topology and weaker kinds of normality. This paper is divided as follows. In Section 2, we review an abstract definition and some basic facts about abstract omega algebras. We support these by presenting three concrete examples. In Section 3, we define a new topology on omega algebra and discuss some of its topolog- ical properties. In Section 4, we provide four different examples of omega topology: the first and fourth examples are from an ordered infinite sets, the second example is from a cyclically ordered infinite set, and the third example is from a finite set. Furthermore, we define a new topology over a semiring in conventional algebra. Finally, we study the rela- tionship between omega topology and weaker kinds of normality in section 5. Throughout this paper, we do not assume T2 in the definition of compactness. We also do not assume regularity in the definition of Lindelöfness. This paper is produced from the PhD thesis of Mr. Mesfer Hayyan Alqahtani in King Abdulaziz University. 2. Preliminaries In this section, we provide an abstract definition and review some basic facts about abstract omega algebras. Furthermore, we support these by presenting concrete examples: one from an ordered infinite set, another from a cyclically ordered infinite set, and a third one from a finite set. For more details, see [11]. Let (G, ◦, e) be an abelian group. Let A be a closed subset of G and e ∈ A. Then (A, ◦, e) is a submonoid of G. Assume that ω is an indeterminate (may belong to A or G, as we will see in Examples 1 and 2). Obviously, in this case ω is no longer an indeterminate. Because the terms are generated from tropical geometry, this indeterminate can be called a tropical indeterminate. Definition 1. [11] We say that Aω = A ∪ {ω} is an omega algebra (in short ω− algebra) over the group G in case Aω is closed under two binary operations, ⊕,⊗ : Aω ×Aω −→ Aω, Then for all a1, a2, a3 ∈ A, the following axioms are satisfied: (i) a1 ⊕ a2 = a1 or a2; (ii) a1 ⊕ ω = a1 = ω ⊕ a1; M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 515 (iii) ω ⊕ ω = ω; (iv) a1 ⊗ a2 = a2 ⊗ a1 ∈ A; (v) (a1 ⊗ a2)⊗ a3 = a1 ⊗ (a2 ⊗ a3); (vi) a1 ⊗ e = a1; (vii) a1 ⊗ ω = ω ⊗ a1 = { ω if ω 6= e a1 if ω = e ; (viii) ω ⊗ ω = ω; (ix) a1 ⊗ (a2 ⊕ a3) = (a1 ⊗ a2)⊕ (a1 ⊗ a3). Remark 1. [11] (1) ⊕ is a pairwise comparison operation such as max, min, inf, sup, up, down, lexico- graphic ordering, or anything else that compairs two elements of Aω. Obviously, it is associative and commutative and the tropical indeterminate ω play the role of the identity. Hence (Aω,⊕, ω) is a commutative monoid. (2) ⊗ is also associative and commutative on Aω, and e plays the role of the multiplica- tive identity of Aω. Hence, (Aω,⊗, e) is also a commutative monoid. (3) The left distributive law (ix) also gives the right distributive law. (4) Every element of Aω is an idempotent under ⊕. (5) Altogether, we write both structures as: Aω = (Aω,⊕,⊗, ω, e). This is an idempotent semiring, which is also called ”dioid” in the literature. Remark 2. [11] ω− algebra can similarly be defined over a commutative monoid, ring, or even a semiring. More generally, one may construct analogously such algebras on other more weaker structures. In this note, we confined ourselves to only ω− algebras over abelian groups and rings. Example 1. [11] Max-plus algebra, min-plus algebra and all such ”so called” algebras are particular cases of the ω− algebra over the ring R or its associated subrings. A simpler example is the following. In the abelian group (Z,+), for any integer m, we have W (m) = {0,m, 2m, · · · }. This is an additive submonoid of (Z,+). Let ω = −∞, a1 ⊕ a2 = max(a1, a2) and a1 ⊗ a2 = a1 + a2, ∀a1, a2 ∈W (m). Then. W (m)−∞ = {W (m)−∞,⊕,⊗,−∞, 0} is −∞ − algebra over the abelian group of integers Z. Hence, we have a sequence of ω− subalgebras W (m) ≥W (2m) ≥ · · · . M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 516 Example 2. [11] A cyclically ordered abelian group. This example is constructed exclu- sively over an abelian group. A cyclically ordered abelian group is more general than that of a linearly ordered abelian group. Every linearly ordered abelian group is cyclically or- dered, but the converse,in general, is not true. The following example is that of a cyclically ordered abelian group, which is not an ordered abelian group [16]. For more details about a cyclically ordered abelian groups. Consider the cyclically ordered abelian group in the form of the unit circle C = {z ∈ C | |z| = 1} . Let W = {0, 1, 2, · · · }. For some θ ∈ [0, 1), define ρx = e2πiθx, where x ∈W, in particular, ρ0 = 1. Set A := {ρx |x ∈W} ⊂ C. Because ρx1ρx2 = ρx1+x2 , ∀x1, x2 ∈W, A is multiplicatively closed. Theorem 1. [11] Aρ0 = A ∪ {ρ0} is an omega algebra with the identical additive and multiplicative identities. This omega algebra contains infinite omega subalgebras. Proof. Define ⊕ on A by ρx1 ⊕ ρx2 = ρx3 where x1, x2, x3 ∈W, with x3 = max(x1, x2) and define ⊗ on A by ρx1 ⊗ ρx2 = ρx1+x2 , where x1, x2 ∈W. Clearly, both operations are associative, and as ρ0 ⊕ ρx1 = ρx1 and ρ0 ⊗ ρx1 = ρx1 , so (A,⊕, ρ0) and (A,⊗, ρ0) are monoids. Finally, ∀x1, x2, x3 ∈W, ρx1 ⊗ (ρx2 ⊕ ρx3) = ρx1 ⊗ ρmax(x2,x3) = ρx1+max(x2,x3) = ρmax(x1+x2,x1+x3) = (ρx1+x2 ⊕ ρx1+x3) = (ρx1 ⊗ ρx2)⊕ (ρx1 ⊗ ρx3). As, ∀x1 ∈W, ρx1 ⊕ ρx1 = ρx1 and ρ0 = 1 we conclude that A = (A,⊕,⊗, 1, 1) is an omega algebra. Finally, consider W (m) = {0,m, 2m, · · · }, where m = 1, 2, · · · . For each m, one can construct an omega subalgebra. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 517 Example 3. [11] A Laxicographic Ordering. Consider the binary linear code of length 2; Z(2) 2 = {00, 01, 10, 11}. Under componentwise addition + and componentwise multiplication ◦, (Z(2) 2 ,+, ◦) is a ring with code-words 0 = 00 and 1 = 11 as additive and multiplicative identities. We define the laxicographic odering on the elements of Z(2) 2 and arrange them as: 00 < 01 < 10 < 11 Let A = {00, 01}. Consider ω = 11. Note that, in this example, ω /∈ A but ω ∈ G. We define addition on Aω = {00, 01, 11} by: a⊕ b = min(a, b). Hence we get the table: ⊕ 00 01 11 00 00 00 00 01 00 01 01 11 00 01 11 . Define multiplication as the boolean sum, namely, 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 1. Hence we get the table: ⊗ 00 01 11 00 00 01 11 01 01 01 11 11 11 11 11 . We conclude that (Aω,⊕, 11) and (Aω,⊗, 00) are the additive and multiplicative monoides. Clearly, this is a simple ω− algebra. 3. Omega topology In this section, we define a new topology on omega algebra and discuss some of its topological properties. Proposition 1. Let (G, ◦, e) be an abelian group and Aω = (Aω,⊕,⊗, ω, e) an ω−algebra over the group G. We define a new topology on Aω is called an omega topology, denoted by τω, as follow: τω = {∅, Aω} ∪ {U ⊆ Aω : ω ∈ U and for any a ∈ U \ {ω}, the multiplicative inverse of exists in U } M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 518 Proof. Condition ∅, Aω ∈ τω is satisfied from the definition of τω. Now let V1, V2 ∈ τω be arbitrary. If either V1 or V2 is empty, then V1∩V2 = ∅ ∈ τω. Assume now, V1 6= ∅ 6= V2. If either V1 or V2 is a whole set Aω, then V1 ∩ V2 = V1 or V2 ∈ τω. So, assume that V1 6= Aω 6= V2, then V1∩V2 ∈ τω, because ω ∈ V1 and ω ∈ V2. Hence ω ∈ V1∩V2. Also for any element (a 6= ω) ∈ V1 ∩ V2, we have a ∈ V1 and a ∈ V2, then a and the multiplicative inverse of a must belong to V1 and V2. Hence a and the multiplicative inverse of a belong to V1 ∩ V2, then V1 ∩ V2 ∈ τω. For the third condition, let Sγ ∈ τω for any γ ∈ I. If Sγ = ∅ for all γ ∈ I, then ⋃ γ∈I Sγ = ∅ ∈ τω. So, assume that some member is non-empty, but since the empty set does not affect any union, assume that, without loss of generality Sγ 6= ∅ for all γ ∈ I. If there exist a γ1 ∈ I where Sγ1 = Aω, then ⋃ γ∈I Sγ = Aω ∈ τω. So, assume now that Sγ 6= Aω for all γ ∈ I, then ⋃ γ∈I Sγ ∈ τω, because ω ∈ Sγ for all γ ∈ I. Hence ω ∈ ⋃ γ∈I Sγ . Also for any (a 6= ω) ∈ ⋃ γ∈I Sγ , there exists γa ∈ I such that a ∈ Sγa , hence a and the multiplicative inverse of a belong to Sγa , then a and the multiplicative inverse of a belong to ⋃ γ∈I Sγ . Hence ⋃ γ∈I Sγ ∈ τω. Therefore, (Aω, τω) is topological space. Corollary 1. If a ∈ Aω \ {ω} has no multiplicative inverse, then Aω is the only open set in (Aω, τω) containing a. Let us denoted for the multiplicative inverse of a ∈ Aω by a−1. If (Aω \ {ω},⊗) is a group, where ω and e are the zero and multiplicative identity elements, respectively, then for any a ∈ Aω \ {ω}, we have a−1 ∈ A . Proposition 2. The omega topological space (Aω, τω) has a base B = {Aω, {ω}, {ω, a, a−1} : a ∈ Aω \ {ω}, which has a multiplicative inverse}. Proof. For the first condition, let B ∈ B be arbitrary, if B = {ω} or Aω, then B ∈ τω is satisfied from the definition of τω. Assuming that B = {ω, a, a−1} for any a ∈ Aω \ {ω} which has a multiplicative inverse, then B ∈ τω, because ω ∈ B, and the element a in B its multiplicative inverse exists in B. Hence, B ⊆ τω. For the second condition, let a ∈ Aω be arbitrary and let U be any open neighborhood of a in Aω. Then we have two cases: Case 1: If a = ω, then we have B = {ω} ∈ B, where ω ∈ B ⊆ U , because the smallest open neighborhood in Aω containing ω is {ω}. Case 2: Let a 6= ω Subcase 2.1: If a has no multiplicative inverse, then there exists B = Aω ∈ B, such that a ∈ B ⊆ U , because the smallest open neighborhood in Aω containing a is Aω. Subcase 2.2: If a has a multiplicative inverse, then there exists B = {ω, a, a−1} ∈ B, such that a ∈ B ⊆ U , because the smallest open neighborhood in Aω containing a is {ω, a, a−1}. Therefore, B is a base for the omega topological space (Aω, τω). Corollary 2. If (Aω \ {ω},⊗) be a group, then the omega topological space (Aω, τω) has a base B = {{ω}, {ω, a, a−1} : a ∈ Aω \ {ω}}. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 519 Proposition 3. If Aω has a finite number of elements, which have a multiplicative in- verses, then the omega topological space (Aω, τω) is second countable. Proof. Suppose that a1, a2, · · · , am, where m ∈ Z+ are the finite number of elements in Aω, which have a multiplicative inverses. Then B = {Aω, {ω}, {ω, a1, a−11 }, · · · , {ω, am, a−1m }} is a countable base for (Aω, τω). Proposition 4. The omega topological space (Aω, τω) is first countable. Proof. Let a ∈ Aω be arbitrary. If a = ω, then B(ω) = {{ω}} is a countable local base at ω. Assume that, a 6= ω. Case 1: If a has a multiplicative inverse, then B(a) = {{ω, a, a−1}} is a countable local base at a. Case 2: If a has no multiplicative inverse, then B(a) = {Aω} is a countable local base at a. Hence for any a ∈ Aω, there exists a countable local base at a. Then (Aω, τω) is first countable. Proposition 5. The omega topological space (Aω, τω) is separable. Proof. There exist {ω} ⊆ Aω, such that for any nonempty U ∈ τω, we have U∩{ω} 6= ∅, because any nonempty open set in (Aω, τω) must be contaning ω. Hence, {ω} = Aω (which means {ω} is a dense subset of Aω). Then Aω has a countable dense subset. Therefore, (Aω, τω) is separable. Let us recall this definition. Definition 2. A topological space X is called hyperconnected if every non-empty open subset is dense in X. Proposition 6. The omega topological space (Aω, τω) is hyperconnected. Proof. If Aω is singleton, then (Aω, τω) is hyperconnected. Suppose that Aω, which has more than one element. Let U be arbitrary non-empty open subset of Aω, then U intersects every non-empty open subset of Aω, because any non-empty open subset of Aω contained ω. Hence U is dense in Aω. Since U was chosen arbitrary, then every non-empty open subset of Aω is dense. Therefore (Aω, τω) is hyperconnected. Since any hyperconnected space is connected and locally connected, then we conclude the following corollaries. Corollary 3. The omega topological space (Aω, τω) is connected. Corollary 4. The omega topological space (Aω, τω) is locally connected. Proposition 7. The omega topological space (Aω, τω) is not T1. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 520 Proof. If Aω is singleton, then it is T1. Assume that Aω, which has more than one element. There exists a, ω ∈ Aω such that a 6= ω. Since any nonempty open set must be containing ω, then we can not find two open sets U and V such that a ∈ U, ω /∈ U, a /∈ V and ω ∈ V. Therefore, (Aω, τω) is not T1. Proposition 8. Let (Aω \ {ω},⊗) be a group, which has more than two elements. Then the omega topological space (Aω, τω) is not T0. Proof. If Aω has only two elements, then it is T0. Assume that Aω, which has more than two elements. If a ∈ Aω is arbitrary, such that a 6= ω and a 6= e, then a 6= a−1 in Aω and any open set containing a must be containing a−1. Hence, (Aω, τω) is not T0. Proposition 9. If Aω, has more than one element, then the omega topological space (Aω, τω) is not regular. Proof. If Aω is singleton then it is regular. There exists K = Aω \ {e, ω} be a closed subset of Aω and ω /∈ K. We can not separated ω and K by any open sets (because any open set in Aω is containing ω). Hence, (Aω, τω) is not regular. Proposition 10. If (Aω \ {ω},⊗) be a group, which has more than two elements, then the omega topological space (Aω, τω) is not normal. Proof. If Aω = {e = ω} or {e, ω}, then it is normal. Assume that Aω has more than two elements, then there exists a ∈ Aω such that a 6= e, a 6= ω and its multiplicative inverse exists in Aω. Then, there exists K = {a, a−1} and H = {e} are two disjoint nonempty closed subsets of Aω, such that we can not separat them by any open sets (because any open set in Aω is containing ω). Hence, (Aω, τω) is not normal. Proposition 11. If (Aω \ {ω},⊗) be a group and A is uncountable infinite set, then the omega topological space (Aω, τω) is not compact (Lindelöf). Proof. There exists {{ω}, {ω, a, a−1} : a ∈ Aω \ {ω}}, which s an open cover of Aω, and has no finite (countable) subcover of Aω. Proposition 12. Let a ∈ Aω \ {ω} has no multiplicative inverse. Then the omega topo- logical space (Aω, τω) is compact. Proof. Let {Cα : α ∈ Λ} be any open cover of Aω. Since a ∈ Aω, then for some β ∈ Λ, there exists Cβ containing a. But Cβ = Aω, because Aω is the only open set containing a. Hence, {Cβ} is a finite subcover of {Cα : α ∈ Λ}, which cover Aω. Therefore, (Aω, τω) is a compact space. Since any compact space is Lindelöf and countably compact, then we conclude the following corollaries. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 521 Corollary 5. If a ∈ Aω \ {ω} has no multiplicative inverse, then the omega topological space (Aω, τω) is Lindelöf. Corollary 6. If a ∈ Aω \ {ω} has no multiplicative inverse, then the omega topological space (Aω, τω) is countably compact. Remark 3. Since every nonempty open set in (Aω, τω) containing ω, then the closure of any nonempty open set is equal Aω. 4. Some of the fundamental properties for different examples on omega topology In this section, we give four different examples of omega topologies. The first and fourth examples are from an ordered infinite set, the second is from a cyclically ordered infinite set, and the third is from a finite set. Furthermore, we define a new topology over semiring in conventional algebra. Example 4. By Example 1, we have (W−∞, τ−∞) which is a topological space, where W = {0, 1, 2, 3, · · · }. If a ∈W \{0} be arbitrary, then a−1 does not exists in (W−∞ \{−∞},⊗), where a−1 is the multiplicative inverse of a. Hence, we have τ−∞ = {W−∞, ∅, {−∞}, {−∞, 0}}. A direct check shows that (W−∞, τ−∞) is a topological space. Proposition 13. The omega topological space (W−∞, τ−∞) is second countable. Proof. There exists an element 2 ∈W−∞ \ {−∞}, which has no multiplicative inverse, then by Proposition 3, (W−∞, τ−∞) is second countable. Corollary 7. The omega topological space (W−∞, τ−∞) is first countable. Corollary 8. The omega topological space (W−∞, τ−∞) is separable. Proposition 14. The omega topological space (W−∞, τ−∞) is not T0. Proof. There exists (2 6= 3) in W−∞. Let U be any open set, which either contains 2 or 3. However, there exists only one open set U = W−∞ containing 2 and 3. Hence, (W−∞, τ−∞) is not T0. Proposition 15. The omega topological space (W−∞, τ−∞) is not regular. Proof. We have a closed set C = W−∞ \ {−∞, 0} and 0 /∈ C, such that for any open sets V1 and V2 containing 0 and C ,respectively, we have V1 ∩V2 6= ∅. Hence, (W−∞, τ−∞) is not regular. Proposition 16. The omega topological space (W−∞, τ−∞) is normal. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 522 Proof. Let K1 and K2 be any two closed sets, where K1∩K2 = ∅ and K1,K2 ⊆W−∞. Since all closed subsets of W−∞ are W−∞, ∅,W−∞ \ {−∞} and W−∞ \ {−∞, 0}, then K1 or K2 is equal ∅. If K1 = ∅, then there exists U1 = ∅ and U2 = W−∞ are open sets and U1 ∩ U2 = ∅ in W−∞, where K1 ⊆ U1 and K2 ⊆ U2. If K2 = ∅, then there exists U1 = ∅ and U2 = W−∞ are open sets and U1 ∩ U2 = ∅ in W−∞, where K2 ⊆ U1 and K1 ⊆ U2. Hence, (W−∞, τ−∞) is normal. Proposition 17. The omega topological space (W−∞, τ−∞) is hyperconnected. Proof. Using the same proof of Proposition 6. Corollary 9. The omega topological space (W−∞, τ−∞) is connected. Corollary 10. The omega topological space (W−∞, τ−∞) is locally connected. Proposition 18. The omega topological space (W−∞, τ−∞) is compact. Proof. There exists an element 2 ∈W−∞ \ {−∞}, which has no multiplicative inverse. Hence, by Proposition 12, (W−∞, τ−∞) is compact. Corollary 11. The omega topological space (W−∞, τ−∞) is countably compact. Corollary 12. The omega topological space (W−∞, τ−∞) is Lindelöf. Example 5. By Example 2, (Aρ0 , τρ0) is a topological space. If ρx ∈ Aρ0\{ρ0} is arbitrary, then ρ−1x does not exists in (Aρ0 \ {ρ0},⊗), where ρ−1x is the multiplicative inverse of ρx (because ρx ⊗ ρ−1x = ρx ⊗ ρx−1 = ρx+x−1 = ρx+(−x) = ρ0 and −x /∈W ). Then we have τρ0 = {Aρ0 , ∅, {ρ0}}. A direct check shows that (Aρ0 , τρ0) is a topological space. Remark 4. The omega topological space (Aρ0 , τρ0) is second countable, first countable, separable, normal, hyperconnected, connected, locally connected, compact, countably compact, Lindelöf, does not satisfy T0 and regular. Proposition 19. The omega topological space (Aρ0 , τρ0) is not homeomorphic to (W−∞, τ−∞). Proof. There exists an open set in W−∞, which consists two elements, and such that an open set does not exists in Aρ0 . Example 6. By Example 3, (A11, τ11) is a topological space. Let a ∈ A11 \ {11} be arbitrary. If a = 00, then the multiplicative inverse of a in A11 is 00. If a = 01, then the multiplicative inverse of a in A11 does not exists. Then we have τ11 = {A11, ∅, {11}, {11, 00}}. A direct check shows that (A11, τ11) is a topological space. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 523 Proposition 20. The omega topological space (A11, τ11) is T0 and does not satisfy T1. Proof. If the space A11 consists of three elements 00, 01 and 11 then we have three cases: Case 1: If 00 6= 01 in A11, then we have {00, 11} an open set, where 00 ∈ {00, 11} and 01 /∈ {00, 11}. Case 2: If 00 6= 11 in A11, then we have {11} an open set, where 11 ∈ {11} and 00 /∈ {11}. Case 3: If 01 6= 11 in A11, then we have {11} an open set, where 11 ∈ {11} and 01 /∈ {11}. Hence, (A11, τ11) is T0. Suppose that (A11, τ11) is T1, then {11} is closed. However, A11 \ {11} = {00, 01} is not open, thus a contradiction. Then (A11, τ11) is not T1. Remark 5. The omega topological space (A11, τ11) is second countable, first countable, separable, not regular, normal, hyperconnected, connected, locally connected, compact, countably compact and Lindelöf. Example 7. In the ring (R,+, ·), (R,+) is an additive submonoid of an abelian group (R,+). Let ω = −∞, a1 ⊕ a2 = max(a1, a2) and a1 ⊗ a2 = a1 + a2,∀a1, a2 ∈ R. Then, R−∞ = (R−∞,⊕,⊗,−∞, 0) is −∞ − algebra over the ring (R,+, ·). Then, using the same proof as that of Proposition 1 (R−∞, τ−∞) is a topological space. Remark 6. The omega topological space (R−∞, τ−∞) is first countable, separable, hyper- connected, connected and locally connected and does not satisfy any of these T0; regular and normal. Example 8. In the ring (R,+, ·), (R,+) is an additive submonoid of an abelian group (R,+). Let ω = +∞, a1 ⊕ a2 = min(a1, a2) and a1 ⊗ a2 = a1 + a2, ∀a1, a2 ∈ R. Then, R+∞ = (R+∞,⊕,⊗,+∞, 0) is +∞ − algebra over the ring (R,+, ·). Then, using the same proof as that of Proposition 1 (R+∞, τ+∞) is a topological space. Proposition 21. The omega topological spaces (R−∞, τ−∞) and (R+∞, τ+∞) are homeo- morphic, where R−∞ is a max−plus algebra and R+∞ is a min−plus algebra. These are special cases of omega algebra. Proof. We have a map h : (R−∞, τ−∞)→ (R+∞, τ+∞) is defined by: h (x1) = { x1 if x1 ∈ R +∞ if x1 = −∞ ; Let x1, x2 ∈ R−∞ be arbitrary. Let h(x1) = h(x2), then x1 = x2. Hence, h is an injective. If x1 ∈ R+∞ is arbitrary, then we have two cases: Case 1: If x1 6= +∞, then there exists x1 ∈ R−∞ \ {−∞}, such that h(x1) = x1. Case 2: If x1 = +∞, then there exists x1 = −∞ ∈ R−∞, such that h(−∞) = +∞. Hence, h is surjective. Let B ∈ τ+∞ be any basic open set. Since (R−∞ \ {−∞},⊗) and (R+∞ \ {+∞},⊗) are groups, then by Problem 2, we have B = {{−∞}, {−∞, c, c−1} : c ∈ R} and M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 524 B = {{+∞}, {+∞, c, c−1} : c ∈ R} are a base for R−∞ and R+∞, respectively. To prove that h is continuous, we have two cases: Case 1: If B = {+∞}, then h−1(B) = h−1({+∞}) = {−∞} ∈ τ−∞. Case 2: If B = {+∞, c, c−1}, then h−1(B) = h−1({+∞, c, c−1}) = {−∞, c, c−1} ∈ τ−∞. Hence, h is continuous. To prove that h−1 is continuous, we have two cases: (since h is one to one and onto, then (h−1)−1(B) = h(B)). Case 1: If B = {−∞}, then (h−1)−1(B) = h(B) = h({−∞}) = {+∞} ∈ τ+∞. Case 2: IfB = {−∞, c, c−1}, then (h−1)−1(B) = h(B) = h({−∞, c, c−1}) = {+∞, c, c−1} ∈ τ+∞. Hence, h−1 is continuous (which means h is open). In conclusion, if h is homeomorphism, then (R−∞, τ−∞) and (R+∞, τ+∞) are homeomor- phic. Theorem 2. Let X be any semiring in conventional algebra, such that e is the zero ele- ment. We define a topology on X is called zero element topology, as follows: τe = {∅, X} ∪ {U ⊆ X : e ∈ U and for any a ∈ U \ {e}, the multiplicative inverse of a exists in U }. Then (X, τe) is a topological space. Proof. Condition ∅, X ∈ τe is satisfied from the definition of τe. Now let U1, U2 ∈ τe be arbitrary. If either U1 = ∅ or U2 = ∅, then U1 ∩U2 = ∅ ∈ τe. Assume, U1 6= ∅ and U2 6= ∅. If either U1 = X or U2 = X, then U1 ∩ U2 = U1 or U2 ∈ τe. So assume, U1 6= X and U2 6= X, then U1 ∩ U2 ∈ τe, because e ∈ U1 and e ∈ U2. Hence e ∈ U1 ∩ U2, also for any element (a 6= e) ∈ U1∩U2, if a ∈ U1 and a ∈ U2, then a and the multiplicative inverse of a must belong to U1 and U2. Hence a and the multiplicative inverse of a belong to U1 ∩U2, then U1 ∩ U2 ∈ τe. For the third condition, let Sγ ∈ τe for any γ ∈ I. If Sγ = ∅ for all γ ∈ I, then ⋃ γ∈I Sγ = ∅ ∈ τe. So, assume that some member is non-empty. However, since the empty set does not affect any union, assume that, without loss of generality Sγ 6= ∅ for all γ ∈ I. If there exists a γ1 ∈ I such that Sγ1 = X, then ⋃ γ∈I Sγ1 = X ∈ τe. So, assume now that Sγ 6= X for all γ ∈ I, then ⋃ γ∈I Sγ ∈ τe, because e ∈ Sγ for all γ ∈ I. Hence e ∈ ⋃ γ∈I Sγ . Also, for any (a 6= e) ∈ ⋃ γ∈I Sγ , there exists γa ∈ I such that a ∈ Sγa . Hence a and the multiplicative inverse of a belong to Sγa . Furthermore, a and the multiplicative inverse of a belong to ⋃ γ∈I Sγ . Hence ⋃ γ∈I Sγ ∈ τe. Therefore, (X, τe) is topological space. Example 9. The space (R,+, ·) is a ring, where 0 and 1 are the zero and unit elements, respectively. Then (R, τ0) is a topological space. The proof is similar to Theorem 2; just replacing X, τe and e by R, τ0 and 0, respectively. Remark 7. The topological space (R, τ0) is first countable, separable, hyperconnected, connected and locally connected, and does not satisfy any of these T0, regular, normal, compact and Lindelöf. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 525 5. Omega topology and other properties Recall that, a subset A of a space X is said to be regularly-open or an open domain if it is the interior of its own closure, see [7]. A set A is said to be a regularly-closed or a closed domain if its complement is an open domain. A subset A of a space X is called a π-closed if it is a finite intersection of closed domain sets, see [17]. A subset A is called a π-open if its complement is a π-closed. If T and T ′ are two topologies on a set X such that T ′ ⊆ T , then T ′ is called the coarser topology than T and T is called the finer. A space X is said to be a π-normal, [3], if any pair of disjoint closed subsets A and B of X, one of which is π-closed, can be separated by two disjoint open subsets. A space X is said to be a almost-normal, [3], if any pair of disjoint closed subsets A and B of X, one of which is closed domain, can be separated by two disjoint open subsets. A space X is said to be a mildly normal, [15], if any pair of disjoint closed domain subsets A and B of X can be separated by two disjoint open subsets. A space (X, T ) is said to be a epi-normal, [5], if there exists a coarser topology T ′ on X such that (X, T ′) is T4- space (normal and T1-space). A space (X, T ) is said to be a epi-mildly normal, [9], if there exists a coarser topology T ′ on X such that (X, T ′) is T2 and almost normal space. Theorem 3. If (Aω\{ω},⊗) be a group has more than one element, then omega topological space (Aω, τω) is π-normal. Proof. Since the only π-closed sets are the ground set Aω and the empty set, then (Aω, τω) is a π-normal. It is clear from the definitions that normal⇒ π − normal⇒ almost normal⇒ mildly normal. (1) By (1) and Theorem 3, we conclude the following Corollaries. Corollary 13. If (Aω \ {ω},⊗) be a group has more than one element, then omega topo- logical space (Aω, τω) is almost normal. Corollary 14. If (Aω \ {ω},⊗) be a group has more than one element, then omega topo- logical space (Aω, τω) is mildly normal. Proposition 22. Any omega topological space (Aω, τω) is not Epi-mildly Normal. Proof. Suppose that, (Aω, τω) is Epi-mildly Normal. Then there exists a coarser topology T ′ on Aω such that (Aω, T ′) is T2 and mildly normal space. Hence (Aω, τω) is T2, thus a contradiction, because (Aω, τω) is not T1 (see Proposition 7). Then (Aω, τω) is not Epi-mildly Normal. Proposition 23. Any omega topological space (Aω, τω) is not Epi-almost Normal. Proof. Using the same proof of Proposition 22. M. Alqahtani, C. Özel, I. Alshammari / Eur. J. Pure Appl. Math, 13 (3) (2020), 513-528 526 Definition 3. Let X be a space. Then: 1) A space X is called a C-normal if there exist a normal space Y and a bijective function f : X → Y such that the restriction function f |A : A → f(A) is a homeo- morphism for each compact subspace A ⊆ X, [2]. 2) A space X is called a CC-normal if there exist a normal space Y and a bijective function f : X → Y such that the restriction function f |A : A → f(A) is a homeo- morphism for each countably compact subspace A ⊆ X.[4]. 3) A spaceX is called a L-normal if there exist a normal space Y and a bijective function f : X → Y such that the restriction function f |A : A → f(A) is a homeomorphism for each lindelöf subspace A ⊆ X.[6]. 4) A space X is called a S- normal if there exist a normal space Y and a bijective function f : X → Y such that the restriction function f |A : A → f(A) is a homeo- morphism for each seprable subspace A ⊆ X.[1]. 5) A space X is called a C-paracompact (C2-paracompact) if there exist a paracompact (Hausdorff paracompact) space Y and a bijective function f : X → Y such that the restriction function f |A : A→ f(A) is a homeomorphism for each compact subspace A ⊆ X.[13]. Theorem 4. If a ∈ Aω \ {ω} has no multiplicative inverse, then omega topological space (Aω, τω) is C-normal. Proof. Let a ∈ Aω \ {ω} has no multiplicative inverse. Let V be any non-empty closed subset of Aω. Then a ∈ V. Suppose not, a /∈ V, then a ∈ Aω \ V. By the definition of τω, Aω \V is not open, thus a contradiction. Hence, a belong to any non-empty closed subsets of Aω. Let K and H be any two disjoint closed subsets of Aω. Then K or H is equal ∅. If K = ∅, then there exists U = ∅ and V = Aω are two disjoint open sets in Aω containing K and H, respectively. If H = ∅, then there exists U = ∅ and V = Aω are two disjoint open sets in Aω containing H and K, respectively. Therefore, (Aω, τω) is normal. Then there exist Y = Aω is a normal space and the identity function id : Aω → Aω is bijective. Let C be any compact subset of (Aω, τω). Then the restriction function id �C : C → f(C) is a homeomorphism. Therefore, (Aω, τω) is a C−normal. Since any normal space is CC-normal, L-normal and S-normal, just by taking X = Y and f to be the identity function. Hence, we conclude the following Theorems. Theorem 5. If a ∈ Aω \ {ω} has no multiplicative inverse, then omega topological space (Aω, τω) is CC-normal. Proof. Using the same proof of Theorem 4. Theorem 6. If a ∈ Aω \ {ω} has no multiplicative inverse, then omega topological space (Aω, τω) is L-normal. REFERENCES 527 Proof. Using the same proof of Theorem 4. Theorem 7. If a ∈ Aω \ {ω} has no multiplicative inverse, then omega topological space (Aω, τω) is S-normal. Proof. Using the same proof of Theorem 4. Example 10. By Example 4, (A−∞, τ−∞) is C-normal, CC-normal, L-normal and S- normal. Theorem 8. If (Aω\{ω},⊗) be a group has more than one element, then omega topological space (Aω, τω) is not S-normal. Proof. From the proposition any separable S-normal must be normal (see [1]) and since (Aω, τω) is separable and not normal (see Proposition 5 and Proposition 10, repectively), then (Aω, τω) is not S-normal. Example 11. By Example 7, (R−∞, τ−∞) is not a S-normal. Theorem 9. Every omega topological space (Aω, τω) is not C2-paracompact. Proof. Since any C2-paracompact Fréchet space is Housdorff, see [13], and (Aω, τω) is First countable not Housdorff space, then (Aω, τω) can not be C2-paracompact. Theorem 10. Let a ∈ Aω \ {ω} has no multiplicative inverse. Then omega topological space (Aω, τω) is not C-paracompact. Proof. Assume that (Aω, τω) is C-paracompact. Let Y be a paracompact space and f : Aω → Y be bijective such that the restriction f �C : C → f(C) is a homeomorphism for all compact subspace C of (Aω, τω). Hence, Aω ≡ Y , since Aω is compact (see Proposition 12). However, Aω is paracompact, thus a contradiction. 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