Open access peer-reviewed chapter

New Topology on Symmetrized Omega Algebra

Written By

Mesfer Alqahtani, Cenap Özel and Ibtesam Alshammari

Submitted: 16 October 2019 Reviewed: 07 May 2020 Published: 13 July 2020

DOI: 10.5772/intechopen.92764

From the Edited Volume

Structure Topology and Symplectic Geometry

Edited by Kamal Shah and Min Lei

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Abstract

The purpose of this paper is to define a new topology called symmetrized omega algebra topology and discuss some of its topological properties. Two different examples from an ordered infinite set of symmetrized omega topology are introduced. Furthermore, we study the relationship between symmetrized omega topology and weaker kinds of normality.

Keywords

  • tropical geometry
  • idempotent semiring
  • topological space
  • topological properties
  • omega algebra and symmetrized omega algebra

1. Introduction

Tropical geometry is the most recent but fast growing branch of mathematical sciences, which is analytically based on idempotent analysis and algebraically on idempotent semirings 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 to the semiring structure, the additive operation of R is used as the multiplication operation. By these choices, both R and R become idempotent semirings. 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. Interestingly, some authors associate R to tropical geometry, while other authors associate R to tropical geometry (see [1, 2, 3, 4]). Omega algebra or “ω 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 minmax and max times algebras, etc., are all subsumed under omega algebra. All these are idempotent semirings, which are also called dioids. In 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 [5]. The aim of this paper is to define a new topology on symmetrized omega algebra, and discus some of its topological properties. Two different examples from an ordered infinite sets of symmetrized omega topology are introduced. Furthermore, we study the relationship between symmetrized omega topology and weaker kinds of normality. Our paper is organized as follows. In Section 2, we review an abstract definition for some basic facts about abstract omega algebras. In addition, we give a brief of symmetrized omega algebra and rules of calculation in omega. In Section 3, we define a new topology on symmetrized omega algebra and discuss some of its topological properties. In Section 4, we provide two different examples of symmetrized omega topology: the first and second examples are from an ordered infinite set. Finally, we study the relationship between symmetrized 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.

The ideas from this paper were taken from the PhD thesis of Mr. Mesfer Hayyan Alqahtani in King Abdulaziz University.

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2. Preliminaries

In this section, we provide an abstract definition for review some basic facts about abstract omega algebra. Furthermore, we also provide a brief of symmetrized omega algebra and rules of calculation in omega. For more details, see [6].

2.1 Omega algebra

Let Ge be an abelian group. Let A be a closed subset of G and eA. Then Ae 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. [6]

We say that Aω=Aω is an omega algebra (in short ωalgebra) over the group G in case Aω is closed under two binary operations,

,:Aω×AωAω,E1

such that a1,a2,a3A, the following axioms are satisfied:

  1. a1a2=a1ora2;

ii. a1ω=a1=ωa1;
iii. ωω=ω;
iv. a1a2=a2a1A;
v. a1a2a3=a1a2a3;
vi. a1e=a1;
vii. a1ω=ωa1=ωifωea1ifω=e;
viii. ωω=ω;
ix. a1a2a3=a1a2a1a3.

Remark 2. [6]

  1. is a pairwise comparison operation such as max, min, inf, sup, up, down, lexicographic ordering, or anything else that compairs two elements of Aω. Obviously, it is associative and commutative and the tropical indeterminate ω plays the role of the identity. Hence Aωω is a commutative monoid.

  2. is also associative and commutative on Aω, and e plays the role of the multiplicative identity of Aω. Hence, Aωe 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=Aωωe. This is an idempotent semiring.

Remark 3. [6] A 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 weaker structures. In this note, we confined ourselves to only ω algebras over abelian groups and rings.

Example 4. [6] Max-plus algebra, min-plus algebra and all such “so called” algebras are particular cases of the ωalgebra 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 Wm=0m2m. This is an additive submonoid of (Z,+). Let ω=,a1a2=maxa1a2 and a1a2=a1+a2,a1,a2Wm. Then,

Wm=Wm0E2

is algebra over the abelian group of integers Z. Hence, we have a sequence of ω subalgebras

WmW2m.

Example 5. [6] Cartesian products of omega algebras. In this example, we explain a construction of an omega algebra from other given omega algebras. Let Giiei:i=1n be abelian groups and Aωiiiωiei:i=1n be a respective family of omega algebras, where ωi are tropical indeterminate. As usual, we define the Cartesian product as

Xω=Aω1××Aωn=a1an:aiAωii=1n.E3

In order to provide a convenient technique to give an additive structure to Xω, we assume that the ntuplesa=a1an,b=b1bnXω are in lexicographic ordering. Then, to define the sum

ab=aorbE4

by using the following rules:

Ifa11b1=a1thenab=a.E5
Ifai=bifor1ikn,andak+1k+1bk+1=ak+1,then,ab=a.E6

Similarly, rules for ab=b can be determined. Multiplication can be to define component wise. Thus,

ab=a11b1annbn.E7

The other rules of the Definition 1 can straightforwardly be verified. Hence, Xωωe, where ω=ω1ωn is the additive identity and e=e1en is the multiplicative identity of Xω, is an omega algebra over the Cartesian product of abelian groups G1××Gn..

2.2 The symmetrized omega algebra

Let Ge be an abelian group and Aωωe an ωalgebra over the group G. Following the method used in constructing integers from the natural numbers, we consider the set of ordered pairs Pω=Aω2 with component wise addition , for all ab, cdPω,

abcd=acbdE8

Because of the four possibilities ab, ad, cd or cb for the result, the addition in (8), is ambiguous. As our goal from constructing the algebra of pairs is the construction of the symmetrized omega algebra of Aω, we are in front of two possibilities: One is to use –for n=2, and define an equivalence relation on the ωalgebra of pairs which is compatible with relevant operations, and the other is to define an equivalence relation on the set Pω that allows the component wise addition to be defined in the quotient set.

First Construction, let be the ordering defined on Aω by the relation

abab=bE9

which gives a total order on Aω and for all aAω, we have ωa. For ab, such that ab=b, we denote by a<b. Under the ordering , rules (5) and (6) defined in Example 5, are satisfied on Pω=Aω2 and so Pω is an ωalgebra under the addition defined in 1 and the component wise multiplication. Let be the relation defined on Pω as follows: for all ab, cdPω

abcdad=bc.E10

Then is reflexive and symmetric but not transitive for Aω contains more than 4 elements. In fact, let a, b, c, dAω such that a<b<c<d, then we have

ad=d=bd=cdandac=cb=bb

which give abdd and ddbc, but there is no relation between ab and bc. As is not an equivalence relation, we cannot use it to obtain the quotient ωalgebra Pω (like the one to obtain integers from the natural numbers).

Definition 6. [6] Let be the equivalence relation close to defined as follows: for all ab, cdPω,

abcdabcdifabandcdab=cdotherwiseE11

In addition to the class element ω¯=ωω¯; for all aAω, with aω, we have three kinds of equivalence classes:

  1. aω¯=abPωb<a, called positive ωelement.

  2. ωa¯=baPωb<a, called negative ωelement.

  3. aa¯ called balanced ωelement.

Unfortunately, the addition defined by (7) and rules (8) and (9) in Example 5 is not compatible with the equivalence relation in Pω, because for aω, ab, ωc, dcPω, such that

aωabωcdc,E12

we have

aωωcabdciffabdc=abE13
and ifabdc=dc,E14

then there is no compatibility. So the omega algebra of pairs cannot produce the symmetrized omega algebra.

Second Construction

Proposition 7. [6]

The addition operation ¯ defined by

ab¯¯cd¯=acbd¯

on the quotient set Pω is well defined and satisfies the axioms i, ii and iii of Definition 1, with the zero class element w¯=ωω¯, except this case aω¯¯ωa¯=ωa¯¯aω¯=aa¯, where aAω\ω does not satisfy the axiom i.

Proposition 8. [6]

  1. The set Pω is closed under the binary multiplication operation ¯ defined as follows: for all ab¯, cd¯Pω;

    ab¯¯cd¯=acbdadbc¯E15

    and satisfies axioms from iv to ix of Definition 1, with the unit class element e¯=eω¯.

  2. In addition, we have for all a, bAω

    1. aω¯¯bω¯=abω¯;

    2. aω¯¯ωb¯=ωab¯;

    3. aω¯¯bb¯=abab¯;

    4. ωa¯¯bb¯=abab¯.

Definition 9. [6] The structure Pω¯¯ω¯e¯ is called the symmetrized ωalgebra over the abelian group G×G and we denote it by Sω.

In the coming sections just for simplicity we will only use and instead the operations ¯ and ¯, respectively.

Remark 10. [6]

  1. Despite the nature of the positive and the negative ωelements, they are not the inverses of each other for the additive operation ¯,

  2. We have three symmetrized ωsubalgebras of Sω,

    Sω+=aω¯aAω,
    Sω=ωa¯aAω,
    Sω0=aa¯aAω.

  3. The three symmetrized ωsubalgebras of Sω are connected by the zero class element ω¯.

  4. The positive ωelements, the negative ωelements and the balanced elements are called signed and denoted by Sω=Sω+Sω, where the zero class ωω¯ corresponds to ω.

2.3 Rules of calculation in omega

Notation 11. [6] LetaAω. Then we admit the following notations:

+a=aω¯,a=ωa¯,a=aa¯.E16

By results in Proposition 7 and Proposition 8 and the above notation, it is easy to verify the rules of calculation in the following proposition:

Proposition 12. [6] For alla,bAω, we have

  1. +a+b=+ab;

  2. +ab=+aifb<abifb>a;aifba

  3. ±ab=±aifb<abifb>a:

  4. ab=ab;

  5. +a+b=+ab;

  6. +ab=ab;

  7. ±ab=ab;

  8. ab=+ab.

From the previous rules, we can notice that the sign of the result in the addition operation follows the greater element in Aω. While in the multiplication operation, the balance sign is the strong one (has priority).

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3. Symmetrized omega topology

In this section, we define a new topology on symmetrized omega algebra and discuss some of its topological properties.

Throughout this paper, we assume that A=.

Proposition 13. Let Sω=Pω¯¯ω¯e¯ be a symmetrized ωalgebra over the abelian group G×G, where Pω=Aω×Aω and A=. We define a new topology on Sω called a symmetrized omega topology, denoted by τω as follow:

τω=Sω{USω:Sω0Uand for any+a,aU, their multiplicative inverses exists in U, whereaAω\ω}.

Proof. Condition ,Sωτω is satisfied from the definition of τω. Now let V1,V2τω be arbitrary. If either V1 or V2 is equal , then V1V2=τω. Assume now, V1V2. If either V1 or V2 is equal Sω, then V1V2=V1orV2τω. So assume that, V1SωV2, then V1V2τω, because Sω0V1 and Sω0V2, hence Sω0V1V2, also for any element +a,aV1V2, where aω, then we have +a,aV1 and +a,aV2, then the multiplicative inverse of +a,a must belong to V1 and V2. Hence, the multiplicative inverse of +a,a belong to V1V2, then V1V2τω. For the third condition let Sγτω for any γΛ. If Sγ= for all γΛ, then γΛSγ=τω. So, assume that some member is nonempty, but since the empty set does not affect any union, we may assume, without loss of generality, that Sγ for all γΛ. If there exist γ1Λ such that Sγ1=Sω, then γΛSγ=Sωτω. So, assume now that SγSω for all γΛ. Then γΛSγτω, because Sω0Sγ for all γΛ. Hence Sω0γΛSγ. Also for any +a,aγΛSγ, where aω, there exists γ1,γ2Λ such that +aSγ1 and aSγ2. Hence the multiplicative inverse of +a,a belong to Sγ1 and Sγ2 respectively, then the multiplicative inverse of +a,a belong to γΛSγ. Hence γΛSγτω.

Therefore, Sωτω is topological space.

Proposition 14. IfSω=Pω¯¯ω¯e¯be a symmetrizedωalgebra over the abelian groupG×G, wherePω=Aω×Aω,andA=. Then an elementahas a multiplicative inverse inAωif and only if the elements+a,ahave a multiplicative inverses inSω.

Proof. Let aAω be arbitrary, which has a multiplicative inverse, denoted by a1, then

+a+a1=aω¯a1ω¯=aa1ωωaωωa1¯=aa1ω¯=aa1ω¯=eω¯=e¯,

then +a1 is a multiplicative inverse of +a in Sω. Also,

aa1=ωa¯ωa1¯=ωωaa1ωa1aω¯=aa1ω¯=aa1ω¯=eω¯=e¯,

then a1 is a multiplicative inverse of a in Sω.

Conversely, let +aSω be arbitrary, which has a multiplicative inverse xy¯, where x,yAω, then we have:

+axy¯=aω¯xy¯=axωyayωx¯=axay¯=axay¯=eω¯=e¯,E17

then ax=e and ay=ω. Hence, x=a1 is the multiplicative of a in Aω.

Let aSω be arbitrary, which has a multiplicative inverse xy¯, where x,yAω, then we have:

axy¯=ωa¯xy¯=ωxayωyax¯=ayax¯=ayax¯=eω¯=e¯,E18

then ay=e and ax=ω. Hence, y=a1 is the multiplicative inverse of a in Aω.

Proposition 15. For anyaSω0, whereωe,thenahas no multiplicative inverse.

Proof. Suppose that, aSω0 has a multiplicative inverse xy¯, where x,yAω, then

axy¯=aa¯xy¯=axayayax¯=axayayax¯=eω¯.E19

Hence, axay=e and ayax=ω, thus a contradiction.

Corollary 16. IfaAω\ωhas no multiplicative inverse, thenSωis the only open set inSωτωcontaining+aanda..

Remark 17.

  1. We denote for any element aSω, by sign.a or signaa, where sign.,signa+;

  2. If a=ω, then a=+a=a;

  3. If a1 is the multiplicative inverse of a in Aω, then +a1 and a1 are the multiplicative inverses of +a and a, respectively in Sω (vice versa);

  4. If a has no multiplicative inverse in Aω, then +a and a have no multiplicative inverses in Sω (vice versa).

Proposition 18. A symmetrized omega topological spaceSωτωhas a base

B=SωSω0Sω0+a+a1Sω0aa1:aAω\ωhasamultiplicative inverse.E20

Proof. For the first condition, let BB be arbitrary. If B=Sω0 or Sω then Bτω (satisfied by the definition of τω). Assuming that, B=Sω0+a+a1orSω0aa1 for any aAω\ω, which has a multiplicative inverse in Aω, then Bτω, because Sω0B, and the elements +aanda in B its multiplicative inverse +a1anda1 respectively, exists in B. Thus Bτω. For the second condition, let signaaSω be arbitrary. Let U be any open neighborhood of signaa in Sω. Then we have three cases:

Case 1: If signa=, then there exists B=Sω0B, such that aBU, because the smallest open neighborhood in Sω containing a is Sω0.

Case 2: If signa=+, where aω (If a=ω, then we have +ω=ω=ω, this is Case 1),

Subcase 2.1: If a has a multiplicative inverse in Aω, then there exists B=Sω0+a+a1B, such that +aBU, because the smallest open neighborhood in Sω containing +a is Sω0+a+a1.

Subcase 2.2: If a has no multiplicative inverse in Aω, then there exists B=Sω, such that +aBU, because the smallest open neighborhood in Sω containing +a is Sω.

Case 3: If signa=, where aω.

Subcase 3.1: If a has a multiplicative inverse in Aω, then there exists B=Sω0aa1B, such that aBU, because the smallest open neighborhood in Sω containing a is Sω0aa1.

Subcase 3.2: If a has no multiplicative inverse in Aω, then there exists B=Sω, such that aBU, because the smallest open neighborhood in Sω containing a is Sω.

Therefore, B is a base for the symmetrized omega topological space Sωτω.

Corollary 19. IfAω\ωbe a group, then the symmetrized omega topological spaceSωτωhas a base,

B=Sω0Sω0+a+a1Sω0aa1:aAω\ω.E21

Corollary 20. LetUAω, thenUτωif and only if for eachsignaaU, there exists basic open setBB, such thatsignaaBU.

Proposition 21. IfAωhas a finite number of elements, which have a multiplicative inverses, then the symmetrized omega topological spaceSωτωis second countable.

Proof. Suppose that a1,a2,,am, where mZ+ are the finite number of elements in Aω, which have a multiplicative inverses. Then.

B=SωSω0Sω0+a1+a11Sω0a1a11Sω0+am+am1Sω0amam1 is a countable base for Sωτω.

Proposition 22. The symmetrized omega topological spaceSωτωis first countable.

Proof. Let signaaSω be arbitrary. Then we have three cases:

Case 1: If signa=, then Ba=Sω0 is a countable local base at a.

Case 2: If signa=+, where aω (If a=ω, then +ω=ω=ω, this is Case 1),

Subcase 2.1: If a has a multiplicative inverse in Aω, then B+a=Sω0+a+a1 is a countable local base at +a.

Subcase 2.2: If a has no multiplicative inverse in Aω, then B+a=Sω is a countable local base at +a.

Case 3: If signa=, where aω,

Subcase 3.1: If a has a multiplicative inverse in Aω, then Ba=Sω0aa1 is a countable local base at a.

Subcase 3.2: If a has no multiplicative inverse in Aω, then Ba=Sω is a countable local base at a. Hence, for any signaaSω, there exists a countable local base at signaa.

Therefore, Sωτω is first countable.

Proposition 23. The symmetrized omega topological spaceSωτωis separable.

Proof. There exists ω=ωω¯Sω, such that for any Uτω, we have Uω, because any open set in Sωτω must be containing Sω0, and ωSω0. Then ω is countable dense subset of Sω. Therefore, Sωτω is separable.

Let us recall this definition.

Definition 24. A topological space X is said to be hyperconnected space if every non-empty open set of X is dense in X or there exists no disjoint non-empty open sets in X.

Proposition 25. The symmetrized omega topological spaceSωτωis hyperconnected.

Proof. If Sω is singleton, then it is hyperconnected. Suppose that Sω, which has more than one element. Since any nonempty open set in Sω is containing Sω0, then Sω has no disjoint nonempty open sets. Hence, Sωτω is hyperconnected.

Since any hyperconnected space is connected and locally connected, then we conclude the following corollaries.

Corollary 26. The symmetrized omega topological spaceSωτωis connected.

Corollary 27. The symmetrized omega topological spaceSωτωis locally connected.

Proposition 28. LetAω\ωbe a group, has more than one element. Then the symmetrized omega topological spaceSωτωis notT0.

Proof. If Aω=ω=e, then Sω=ω is singleton, we are done (because some of omega algebra, has ω=e). Suppose that Aω has more than one element. Let aω, then there exist aω in Sω. Let U be any open set in Sω, containing either a or ω, by the definition of τω we have Sω0U, but ω,aSω0. Then there is no open set containing only ω or a. Hence, Sωτω is not T0.

Proposition 29. IfAω\ωbe a group, has more than one element, then the symmetrized omega topological spaceSωτωis not regular.

Proof. There exists K=Sω\Sω0 is a closed subset of Sω and there exists aω, such that aK. We cannot separate a, and K by any open sets (because any open sets in Sω is containing Sω0, where aSω0). Therefore, Sωτω is not regular.

Proposition 30. IfAω\ωbe a group, has more than one element, then the symmetrized omega topological spaceSωτωis not normal.

Proof. If Aω=ω, then Sω=ω is singleton, we are done (because some of omega algebra, we have ω=e). Suppose that Aω has more than one element. Let aAω\ω. Then we have two cases:

Case 1: If a=e, then we have K=+e, and H=e are two disjoint closed subsets of Sω, such that we cannot separate them by any open sets (because any nonempty open sets in Sω is containing Sω0).

Case 2: If ae, then we have K=+a+a1, and H=aa1 are two disjoint closed subsets of Sω, such that we cannot separate them by any open sets (because any nonempty open sets in Sω is containing Sω0). Therefore, Sωτω is not normal.

Proposition 31. IfaAω\ωhas no multiplicative inverse, then the symmetrized omega topological spaceSωτωis normal.

Proof. Suppose that, V be any non-empty closed subset of Sω. Then +aV. Suppose not, +aV, then +aSω\V. By the definition of τω,Sω\V is not open, thus a contradiction. Hence, +a belong to any non-empty closed subsets of Sω. Let K and H be any two disjoint closed subsets of Sω. Then K or H is equal . If K=, then there exists U= and V=Sω are two disjoint open sets in Sω containing K and H, respectively. If H=, then there exists U= and V=Sω are two disjoint open sets in Sω containing H and K, respectively. Therefore, Sωτω is normal.

Proposition 32. IfAω\ωbe a group andAis uncountable infinite set, then the symmetrized omega topological spaceSωτωis not compact (Lindelöf).

Proof. There exists Sω0Sω0+a+a1Sω0aa1:aAω\ω, which is an open cover of Sω, and has no finite (countable) subcover of Sω.

Proposition 33. LetaAω\ωhas no multiplicative inverse. Then the symmetrized omega topological spaceSωτωis compact.

Proof. Let Cα:αΛ be any open cover of Sω. Since +aSω, then for some βΛ, there exists Cβ containing +a. But Cβ=Sω, because Sω is the only open set containing +a. Hence, Cβ is a finite subcover of Cα:αΛ, which cover Sω. Therefore Sωτω is a compact space.

Since any compact space is Lindelo¨f and countably compact, then we conclude the following corollaries.

Corollary 34. IfaAω\ωhas no multiplicative inverse, then the symmetrized omega topological spaceSωτωis Lindelo¨f.

Corollary 35. IfaAω\ωhas no multiplicative inverse, then the symmetrized omega topological spaceSωτωis countably compact.

Remark 36. Since every nonempty open sets in Sωτω contains Sω0. Then the closure of any nonempty open sets is equal Sω.

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4. Some of the fundamental properties for different examples on symmetrized omega topology

In this section, we give two different examples of symmetrized omega topologies. The examples are from an ordered infinite set.

Example 37. By Example 4, we set W=0,1,2,3. Then Sτ, which is topological space, where S=P¯¯¯0¯ be a symmetrized algebra over the abelian group Z×Z and P=W×W. Let aW\0 be arbitrary. Then +a1 and a1 are not exists in S, where +a1 and a1 are the multiplicative inverses of +a and a in S respectively (because a in W has no multiplicative inverse). If a=0, then +01=+0 and 01=0 (because the multiplicative inverse of 0 in W is 0, that is 01=0). Hence,

τ=SS0S0+0S00S0+00.E22

A direct check shows that Sτ is a topological space.

Proposition 38. The symmetrized omega topological spaceSτis Second countable.

Proof. There exists only one element 0W, which has a multiplicative inverse, then by Proposition 21, Sτ is second countable.

Since any second countable space is first countable and separable, then we conclude the following corollaries.

Corollary 39. The symmetrized omega topological spaceSτis first countable.

Corollary 40. The symmetrized omega topological spaceSτis separable.

Proposition 41. The symmetrized omega topological spaceSτis notT0.

Proof. There exists +2+3 in S. Let U be any open set, which either containing +2 or +3. However, there exists only one open set U=S containing +2,+3. Hence, Sτ is not T0.

Proposition 42. The symmetrized omega topological spaceSτis not regular.

Proof. There exists a closed set K=S\S0+0 and +0K, such that +0 and K cannot separate by any two disjoint open sets. Hence, Sτ is not regular.

Proposition 43. The symmetrized omega topological spaceSτis normal.

Proof. There exists an element 2W\, which has no multiplicative inverse, then by Proposition 31, Sτ is a normal space.

Proposition 44. The symmetrized omega topological spaceSτis hyperconnected.

Proof. Since any nonempty open set in S is containing S0, then S has no disjoint nonempty open sets. Hence, Sτ is hyperconnected.

Since any hyperconnected space is connected and locally connected, then we conclude the following corollaries.

Corollary 45. The symmetrized omega topological spaceSτis connected.

Corollary 46. The symmetrized omega topological spaceSτis locally connected.

Proposition 47. The symmetrized omega topological spaceSτis compact.

Proof. There exists an element 2W\, which has no multiplicative inverse. Hence by Proposition 33, Sτ is compact.

Since any compact space is Lindelo¨f and countably compact, then we conclude the following corollaries.

Corollary 48. The symmetrized omega topological spaceSτis countably compact.

Corollary 49. The symmetrized omega topological spaceSτis Lindelöf.

Example 50. In the ring R+, we have R+ is an additive submonoid of an abelian group R+. Let ω=,ab=maxab and ab=a+b,a,bR. Then R=R0 is algebra over the ring R+. We have S=P¯¯¯0¯ be a symmetrized algebra over the abelian group R×R and P=R×R. Then, using the same proof as that Proposition 13. Therefore, Sτ is a topological space.

Remark 51. The symmetrized omega topological space Sτ is first countable, separable, hyperconnected, connected and locally connected and does not satisfy any of these T0, regular, normal, Lindelo¨f and compact.

Example 52. In the ring R+, we have R+ is an additive submonoid of an abelian group R+. Let ω=+,ab=minab and ab=a+b,a,bR. Then, R+=R++0 is + algebra over the ring R+. We have S+=P+¯¯+¯0¯ be a symmetrized +algebra over the abelian group R×R and P+=R+×R+. Then, using the same proof as that Proposition 13. Therefore, S+τ+ is a topological space.

Proposition 53. The symmetrized omega topological spacesSτandS+τ+are homeomorphic.

Proof. There exists a map h:SτS+τ+ is defined by:

hsignaa=signaaifaRsign+ifsignaa=sign;E23

Let signaa,signbbS be arbitrary. Let hsignaa=hsignbb, then signaa=signbb. Hence, h is an injective. Let signaaS+ is arbitrary, then there exists a signaaS, such that hsignaa=signaa. Hence, h is surjective.

Let Bτ+ be any basic open set. By Proposition 18, we have B=S0S0+a+a1S0aa1:aR and B=S+0S+0+a+a1S+0aa1:aR are a base for R and R+, respectively.

To prove that h is continuous, we have three cases:

Case 1: If B=S+0, then h1B=h1S+0=S0τ.

Case 2: If B=S+0+a+a1, then h1B=h1S+0+a+a1=S0+a+a1τ.

Case 3: If B=S+0aa1, then h1B=h1S+0aa1=S0aa1τ. Hence, h is continuous.

To prove that h1 is continuous, we have three cases: (since h is one to one and onto, then h11B=hB).

Case 1: If B=S0, then h11B=hB=hS0=S+0τ+.

Case 2: If B=S0+a+a1, then h11B=hB=hS0+a+a1=S+0+a+a1τ+.

Case 3: If B=S0aa1, then h11B=hB=hS0aa1=S+0aa1τ+. Hence h1 is continuous (which means h is open).

Therefore, h is homeomorphism, then Sτ and S+τ+ are homeomorphic.

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5. Symmetrized 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 [8]). 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 π-normal [9] 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 almost-normal [9] if any pair of disjoint closed subsets A and B of X, one of which is a closed domain, can be separated by two disjoint open subsets. A space X is mildly normal [10] if any pair of disjoint closed domain subsets A and B of X can be separated by two disjoint open subsets. A space XT is epi-mildly normal [11] if there exists a coarser topology T on X such that XT is T2 and mildly normal space. A space XT is epi-almost normal [12] if there exists a coarser topology T on X such that XT is T2 and almost normal space.

Theorem 54. IfAω\ωbe a group has more than one element, then the symmetrized omega topological spaceSωτωisπ-normal.

Proof. Since the only π-closed sets are the ground set Sω and the empty set, then Sωτω is a π-normal.

It is clear from the definitions that

normalπnormalalmost normalmildly normal.E24

By (24) and Theorem 54, we conclude the following Corollaries.

Corollary 55. IfAω\ωbe a group has more than one element, then the symmetrized omega topological spaceSωτωis almost normal.

Corollary 56. IfAω\ωbe a group has more than one element, then the symmetrized omega topological spaceSωτωis mildly normal.

If Aω\ω be a group has more than one element, then Sωτω is not T0(see Proposition 28), we have the following Propositions:

Proposition 57. IfAω\ωbe a group has more than one element, then the symmetrized omega topological spaceSωτωis not Epi-mildly Normal.

Proof. Suppose that, Sωτω is Epi-mildly Normal. Then there exists a coarser topology T on Sω such that SωT is T2 and mildly normal space. Hence Sωτω is T2, thus a contradiction. Then Sωτω is not Epi-mildly Normal.

Proposition 58. IfAω\ωbe a group has more than one element, then the symmetrized omega topological spaceSωτωis not Epi-almost Normal.

Proof. Using the same proof of Proposition 57.

Definition 59. 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:XY such that the restriction function fA:AfA is a homeomorphism for each compact subspace AX, [13].

  2. A space X is called a CC-normal if there exists a normal space Y and a bijective function f:XY such that the restriction function fA:AfA is a homeomorphism for each countably compact subspace AX. [14].

  3. A space X is called an L-normal if there exist a normal space Y and a bijective function f:XY such that the restriction function fA:AfA is a homeomorphism for each lindelöf subspace AX, [15].

  4. A space X is called an S- normal if there exist a normal space Y and a bijective function f:XY such that the restriction function fA:AfA is a homeomorphism for each separable subspace AX, [16].

  5. A space X is called a C-paracompact if there exist a paracompact space Y and a bijective function f:XY such that the restriction function fA:AfA is a homeomorphism for each compact subspace AX, [17].

  6. A space X is called a C2-paracompact if there exist a Hausdorff paracompact space Y and a bijective function f:XY such that the restriction function fA:AfA is a homeomorphism for each compact subspace AX, [17].

Proposition 60. IfaAω\ωhas no multiplicative inverse, then the symmetrized omega topological spaceSωτωisC-normal.

Proof. By Proposition 31, Sω is a normal space. Then there exist Y=Sω is a normal space and the identity function id:SωSω is bijective. Let C be any compact subset of Sωτω. Then the restriction function idC:CfC is a homeomorphism. Therefore, Sωτω is a Cnormal.

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 Propositions.

Proposition 61. IfaAω\ωhas no multiplicative inverse, then the symmetrized omega topological spaceSωτωisCC-normal.

Proof. Using the same proof of Proposition 60.

Proposition 62. IfaAω\ωhas no multiplicative inverse, then the symmetrized omega topological spaceSωτωisL-normal.

Proof. Using the same proof of Proposition 60.

Proposition 63. IfaAω\ωhas no multiplicative inverse, then the symmetrized omega topological spaceSωτωisS-normal.

Proof. Using the same proof of Proposition 60.

Example 64. By Example 37, Sτ is C-normal, CC-normal, L-normal and S-normal.

Theorem 65. IfAω\ωbe a group has more than one element, then the symmetrized omega topological spaceSωτωis notS-normal.

Proof. From the proposition any separable S-normal must be normal (see [16]) and since Sωτω is separable and not normal (see Propositions 30, 23, respectively), then Sωτω is not S-normal.

Example 66. By Example 50, Sτ is not a S-normal.

Theorem 67. The symmetrized omega topological spaceSωτωis notC2-paracompact.

Proof. Since any C2-paracompact Frechet space is Hausdorff (see [17]) and Sωτω is First countable and not a Hausdorff space, Sωτω cannot be C2-paracompact.

Theorem 68. LetaAω\ωhas no multiplicative inverse. Then the symmetrized omega topological spaceSωτωis notC-paracompact.

Proof. Assume that Sωτω is C-paracompact. Let Y be a paracompact space and f:SωY be bijective such that the restriction fC:CfC is a homeomorphism for all compact subspace C of Sωτω. Hence, SωY, since Sω is compact (see Proposition 33). However, Sω is paracompact, thus a contradiction. Because any paracompact space is Hausdorff space and Sω is not a Hausdorff space. Therefore, Sωτω is not a C-paracompact.

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Written By

Mesfer Alqahtani, Cenap Özel and Ibtesam Alshammari

Submitted: 16 October 2019 Reviewed: 07 May 2020 Published: 13 July 2020