SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v16iSP4.ICAMS106research articleOn Microg^π- Closed Sets in Micro Topological SpacesVijayalakshmiRvijayalakshmir218@gmail.com1ThiripurasundariV2Research Scholar, PG and Research Department of Mathematics, Sri S. Ramasamy Naidu Memorial College (Autonomous), Sattur, Affiliated to Madurai Kamaraj UniversityMadurai, Tamil NaduIndiaAssistant Professor, PG and Research Department of Mathematics, Sri S. Ramasamy Naidu Memorial College (Autonomous), Sattur, Affiliated to Madurai Kamaraj UniversityMadurai, Tamil NaduIndia2512202316SP41812102023221120232023
<bold id="s-08492820ea20">Abstract</bold>

Objective: To introduce a new class of sets namely Micro-g^π -closed (briefly μg^π -closed) sets in Micro topological spaces. Methods:The study investigated the concepts of μg^π -closed sets and brief study of μg^π -closed set was made. Findings: We derived the inter-relations between μg^π -closed sets with already existing Micro closed sets in Micro topological spaces and found some of its basic properties. Novelty:Application of μg^π -sets to introduce a new class of space namely μT^1/2 -space.

Keywords: Micro open set, Micro-gs open set, μg^π-closed set, μg^π- open set, μT^1/2-space

Micro topology was introduced by Sakkraiveeranan Chandrasekar1 and he also introduced the concepts of Micro pre-open and Micro semi-open sets. Recently, we initiated the concept of g^π- closed sets in topological spaces2 and also studied its properties. In this paper, we have introduced a new class of Micro closed sets called Micro g^π -closed sets and its properties are studied in Micro topological spaces. Further, we have derived relations between Micro g^π- closed sets with already existing various Micro closed sets. Later, we have defined and analysed μT^1/2- space.

<bold id="s-69c1cebc847b">Preliminaries</bold>

In this paper, (Ω,N,M) denote the micro topological spaces, where N=τR(X), M=μR(X) and MTS denote micro topological space appropriately. For a subset P of a space, clμ(P) and intμ(P) denotes the closure of P and the interior of P respectively.

Definition 2.1. 1 Let (U,τR(X)) be a Nano topological space. Then μRX={N∪N'∩μ:N,N'∈τRX} and μ≠τR(X) and μRX satisfies the following axioms:

1. U and φ are in μR(X)

2. The union of the elements of any sub-collection of μR(X) is in μR(X)

3. The intersection of the elements of any finite sub-collection of μR(X) is in μR(X)

Then, μRX is called the Micro topology on U with respect to X. The triplet (U,τRX,μRX) is Micro topological space and the elements of μRX are Micro open sets and the complement of a Micro open set is called a Micro closed set.

Definition 2.2. Let (Ω,N,M) be a Micro topological space. A subset of A is

i. Micro-sg-closed, if sclμ(A)⊆L, A⊆L and L is Micro- s- open in U. 2

ii. Micro-gs-closed, if sclμ(A)⊆L, A⊆L and L is Micro-open in U. 2

iii. Micro-αg-closed, if αclμ(A)⊆L, A⊆L and L is Micro- open in U. 3

iv. Micro-gα-closed, if αclμ(A)⊆L, A⊆L and L is Micro- α- open in U. 3

v. Micro-g*-closed, if clμ(A)⊆L, A⊆L and L is Micro- g-open in U. 4

vi. Micro-g-closed, if clμ(A)⊆L, A⊆L and L is Micro- open in U. 5

vii. Micro-ψ-closed, if sclμ(A)⊆L, A⊆L and L is Micro-sg-open in U. 6

In this section, we have derived the characteristics of Micro- g^π (shortly μg^π) closed set and its inter-relations with existing other Micro closed sets.

Definition 3.1. Consider (Ω,N,M) as MTS and P⊆Ω . Then P is defined as μg^π - Closed set if πclμ(P)⊆Lwhenever P⊆L and L is Micro-gs -open in Ω.

Theorem 3.2. Every Micro- π- closed set is μg^π -closed set but not conversely.

Proof. Consider a Micro-π-closed set P in Ω such that P⊆L where L is a μgs - open. Therefore, P=πclμ(P)⊆L. Thus, P is μg^π -closed set.

Example 3.3. Consider Ω={u,v,w,x} with Ω/R={{u},{w},{v,x}}. Let X={u,v}⊆Ω, then N={Ω,φ,{u},{u,v,x},{v,x}}. If μ={w}, then M={Ω,φ,{u},{w},{u,w},{v,x},{v,w,x},{u,v,x}. Though the set P={u,w,x} is μg^π-closed it is not Micro-π -closed.

Theorem 3.4. Every μg^π - closed set is Micro-g -semi closed set but not conversely.

Proof. Consider a μg^π -closed set P in Ω such that P⊆L where L is Micro-gs- open. Since πclμ(P)⊆L, sclμ(P)⊆πclμ(P)⊆L. Hence, P is Micro-gs - closed set.

Example 3.5. Let Ω={u,v,w,x} with Ω/R={{u},{w},{v,x}}. Let X={v,x}⊆Ω, then N={Ω,φ,{v,x}. If μ={v}, then M={Ω,φ,{v},{v,x}. Though the set P={u} is Micro-gs – closed, it is not μg^π -closed.

Theorem 3.6. Every μg^π - closed set is Micro- g -closed set but not conversely.

Proof. Consider a μg^π -closed set P in Ω such that P⊆L where L is a μg -semi open. We know that every Microopen set is Micro-g-semi open, so πclμ(P)⊆L. Therefore, clμ(P)⊆πclμ(P)⊆L. Thus, P is Micro-g-closed set.

Example 3.7. Let Ω={u,v,w,x}, Ω/R={{u},{w},{v,x}}. Let X={u,v}⊆Ω, then N={U,φ,{u},{u,v,x},{v,x}}. If μ={w}, then the M={Ω,φ,{u},{w},{u,w},{v,x},{v,w,x},{u,v,x}. Though the set P={u,v} is Micro-g –closed, it is not μg^π -closed.

Theorem 3.8. Every μg^π - closed set is Micro-g* -closed set but not conversely.

Proof. Consider a μg^π -closed set in Ω such that P⊆L where L is a μg -semi open. We know that every Micro-g-open set Micro-gs -open, so πclμ(P)⊆L. Therefore, clμ(P)⊆πclμ(P)⊆L. Thus, P is μg* -closed set.

Example 3.9. Let Ω={u,v,w,x}with Ω/R={{w},{x},{u,v}}. Let X={w}⊆Ω, then N={Ω,φ,{w}}. If P={v} , then M={Ω,φ,v,w,{v,w}}. Then A={u,v,x} is Micro-g* -closed but it is not μg^π –closed.

Theorem 3.10. Every μg^π - closed set is Micro-sg -closed set but not conversely.

Proof. Consider a μg^π -closed set P in Ω such that P⊆L where L is a μg -semi open. We know that every Micro-s-open set is Micro-gs-open, so πclμ(P)⊆L. Then, sclμ(P)⊆πclμ(P)⊆L. Thus, P is Micro-sg -closed set.

Example 3.11. Let Ω={u,v,w,x} withΩ/R={{w},{x},{u,v}}. Let X={w}⊆Ω, then N={Ω,φ,{w}}. If μ={w}, then M={Ω,φ,v,w,{v,w}}. Though the set P={l} is Micro-sg –closed, it is not μg^π -closed.

Theorem 3.12. Every μg^π - closed set is Micro-αg -closed set but not conversely.

Proof. Consider μg^π -closed set in Ω such that P⊆L where L is a μg - semi open. We know that every Micro-open set is Micro-gs -open, so πclμ(P)⊆L. Then, sclμ(P)⊆πclμ(P)⊆L. Thus, P is Micro-αg -closed set.

Example 3.13. Let Ω=u,v,w,x with Ω/R={{u},{v},{w,x}}. Let ={u,w}⊆Ω, then τR(X)={Ω,φ,{u},{u,w,x},{w,x}}. If ={w}, then M={Ω,φ,{u},{w},{u,w},{v,x},{v,w,x},{u,v,x}. Though the set P={u,v,x} is Micro-αg –closed, it is not Micro-g^π -closed.

Remark 3.14. The following Implication diagram shows that the inter-relations with some other existing sets.

Theorem 3.15. The union of two μg^π -closed subset of (Ω,N,M) is also a μg^π -closed.

Proof. Consider two μg^π -closed sets P and Q in Ω,N,M. L is Micro g -semi open sets in Ω containing P∪Q. Then, πclμ(P∪Q)⊆πclμ(P)∪πclμ(P)⊆L. Thus, P∪Qis μg^π closed.

Remark 3.16. The intersection of two μg^π -closed sets in (Ω,N,M) need not be μg^π -closed in (Ω,N,M)

Example 3.17. Consider Ω={w,x,y,z} with Ω/R=w,y,x,z. Let X={w,x}⊆Ω, then N={Ω,φ,{w},{w,x,z},{x,z}}. If μ={y} then M={Ω,φ,{w},{y},{w,y},{x,z},{x.y,z},{w,x,z}}. Then the sets {w,y,z} and {w,x,z} are μg^π -Closed sets but their intersection {z} is not Micro- g^π -Closed.

Theorem 3.18. If P is μg^π - closed in Ω and P⊆Q⊆πclμ(P), then Q is also μg^π closed in Ω.

Proof. Consider a μg^π- closed set P in X here with P⊆Q⊆πclμ(P)⊆L. Suppose that L is μgs-open of X with Q⊆L, then P⊆L⇒πclμ(P)⊆L. So πclμ(Q)⊆L and Q is μg^π - closed in Ω.

Theorem 3.19. If P is a μg^π - closed set of Ω if and only if πclμP-P does not consists of any non-empty Micro-gs - closed.

Proof. Suppose there exist a non-empty Micro-gs- closed set V of Ω such that V⊆πclμP-P, then V⊆πclμ(P). Since P is μg^π - closed and Ω-V Micro-gs-open, πclμP⊆Ω-V. This implies V⊆Ω-πclμP. So V⊆(πclμP-P)∩(Ω-πclμ(P))⊆πclμP∩(Ω-πclμ(P))=φ⇒V=φ.

Conversely, assume that πclμP-P consists of no non-empty Micro-gs- closed such that P⊆L where L is Micro-gs - open set. Assume that πclμis not in L. Then πclμ(P)∩Lc is a non-empty Micro-gs-closed set in πclμP-P which is a contradiction. Then πclμ(P)⊆L and hence P is μg^π - closed set.

Theorem 3.20. The intersection of Micro-gs - closed and μg^π –closed is always μg^π -Closed.

Proof. Consider μg^π –closed set P and μgs – closed set V. This implies U is μgs – open set with P∩V⊆U. Then, P⊆U∪Vc is μgs -open. Since P is μg^π –closed, πclμ(P)⊆U∪Vc⇒πclμ(P)⊆V⊆U. Thus, πclμ(P∩V)⊆πclμ(P)∩πclμ(V)⊆πclμ(P)∩V⊆U. Hence, P∩Visμg^π -closed.

Definition 3.21. Let P⊆Y⊆Ω . Then P is μg^π – closed with relative to Y if πclμY(A)⊆Uwhere A⊆U and U is Micro- g – semi open in Y.

Theorem 3.22. Let P⊆Y⊆Ω and suppose that P is μg^π closed in X. Then P is μg^π - closed with relative to Y.

Proof. Let us assume that P⊆Y∩Z where Z is μgs-open in X. P⊆Z⇒πclμ(P)⊆Z. This implies that πclμ(P)∩Y⊆Z∩Y. Thus, P is μg^π – closed relative to Y.

Definition 3.23. A subset P in Ω is defined as μg^π -open in Ω if Ω-P is μg^π closed in Ω.

Theorem 3.24. If πintμ(P)⊆Q⊆P and P is μg^π -open in Ω, then Q is μg^π -open in Ω.

Proof. Suppose that πintμ(P)⊆Q⊆P and P is μg^π -open in Ω. Then Ω\P⊆Ω\Q⊆πclμ(Ω\P). Since Ω\P is μg^π closed in Ω implies that Ω\Q is μg^π closed in Ω. Hence, Q is μg^π in Ω.

Theorem 3.25. Consider a MTS Ω and S,T⊆Ω. If S is μg^π -open and πintμ(T)⊆S, then S∩T is μg^π -open.

Proof. Given T is μg^π-open and πintμ(T)⊆S, πintμT⊆S∩T⊆T. Hence, S∩T is μg^π -open.

Theorem 3.26. A set P is μg^π -open in Ω if and only if V⊆πintμPwhenever V is Micro-gs - closed in Ω and V⊆P.

Proof. Suppose V⊆πintμP,V is μgs -closed in Ωand V⊆P. Let Ω-P⊆G where G is μgs- open in Ω. So that G⊆Ω-P and Ω-G⊆πintμ(P). Thus, Ω-P is μg^π -closed in Ω. Hence, P is μg^π -open in Ω.

Conversely, suppose that P is μg^π -open, V⊆P and V is μgs - closed in Ω. Then Ω-V is Micro-gs-open and Ω-P⊆Ω-V. But πintμΩ-P=Ω-πintP. Hence V⊆πintμ(P).

Theorem 3.27. If P is μg^π - open in Ω, then U=Ω when U is Micro-gs -open and πintμ(P)⊆Pc⊆U.

Proof. Given P is a μg^π open and U is a Micro-gs - open, πintμ(P)∪Pc⊆U. This gives Uc⊆(X-πintμ(P))∩P=πintμ(Pc)-Pc. Since Uc is μgs -closed and P is μg^π - open. We have Uc=φ. Thus, U=Ω.

Definition 3.28. Let (Ω,N,M) be a MTS. Then Ω is said to be μT^1/2 -space if every μg^π -closed set in Ω is Micro-π -closed in Ω.

Theorem 3.29. For a MTS (Ω,N,M) the following conditions are equivalent.

(i) (Ω,N,M) is a μT^1/2 -space

(ii) Every singleton set {p}is either Micro-gs - closed or Micro-π -open.

Proof. (i) =) (ii) Take p∈Ω. If {p}is not a Micro-gs- closed set of (Ω,N,M). Then Ω-{p} is not a Micro-gs- open set. Thus, Ω-{p} is an μg^π -Closed set of (Ω,N,M). Since (Ω,N,M) is μT^1/2 -space, Ω-{p} is Micro-π -closed set of (Ω,N,M) , That is {p}is Micro-π -open set of (Ω,N,M).

(ii) =) (i) Let P be an μg^π -Closed set of (Ω,N,M). Let p∈πclμ(P). By (ii), {p} is either Micro-gs -closed or Micro-π -open.

Case(i): If {p} is Micro-gs -closed and p∉P. Then πclμ(P)-P contains a non-empty Micro-gs- closed set. This contradicts Theorem 3.19 as P is a Micro-gs - closed set. Therefore, p∈P.

Case(ii): Consider a Micro-π-open set p. Then Ω-p is Micro-π -closed. If p∉P, then P⊆Ω-p. Since pϵπclμ(P), we have p∈Ω-{p}, which is a contradiction. Hence, p∈P.

So in both cases we have πclμ(P)⊆P. Trivially P⊆πclμ(P). Therefore, P=πclμ(P) or equivalently P is Micro-π -closed. Hence, (Ω,N,M) is a μT^1/2 -space.

<bold id="s-22e7a7cfa6ea">Conclusion</bold>

A new class of sets called μg^π-closed sets have been introduced and some of their properties have been studied. Also, μT^1/2-spaces is presented and its properties are analyzed. Furthermore, μg^π-sets can be used to derive a new class of continuity, closed maps, homeomorphism.

<bold id="s-7b9c31a36fcc">Declaration</bold>

Presented in ‘International Conference on Applied Mathematical Sciences’ (ICAMS 2022) during 21 & 22 Dec. 2022, organized virtually by the Department of Mathematics, JJ College of Arts and Science, Pudukkottai, Tamil Nadu, India. The Organizers claim the peer review responsibility.

ReferencesChandrasekarSOn Micro topological spacesThiripurasundariVVijayalakshmiRA New type of Generalized Closed Set in Topological SpacesIbrahimH ZMicro β-open sets in Micro topologyIbrahimH ZOn Micro T1/2 -spaceSandhiyaSBalamaniNMicro g* -closed sets in Micro Topological SpacesSowmiyaTBalamaniNMicro ψ-Closed Sets in Micro Topological Spaces