EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6135 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Study on (c,d) IF −Q Uniform Ir∗ Centred Structure Compactification S. Thirukumaran1, G. K. Revathi∗1 1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Chennai, Chennai-127, Tamilnadu Abstract. Compactification is one of the novel extensions in topological space. Nets and filters are used to study the detailed characterization of compactness and convergence in topological spaces. The major framework of this article delves into (c,d) IF −Q uniform Ir∗ centred struc- ture compactification. An innovative space that integrates a (c,d) IF −Q uniform topological space with (c,d) IF −Q uniform Ir∗ space. It explains about the irreducibility in (c,d) IF −Q uniform topological space. Also combines with (c,d) IF −Q uniform centred system that deals the intersection of open sets. This study also involves (c,d) IF −Q uniform Ir∗ centred structure filters and (c,d) IF −Q uniform Ir∗ centred structure nets which explains a detailed analysis on sequences and its convergence in (c,d) IF −Q uniform topological space. 2020 Mathematics Subject Classifications: 54A40, 03E72 Key Words and Phrases: (c,d) IF −Q uniform Ir∗ structure space, (c,d) IF −Q uniform Ir∗ centred structure space, (c,d) IF −Q uniform Ir∗ centred structure filter and (c,d) IF −Q uniform Ir∗ centred structure net 1. Introduction L. Zadeh in 1965, [1] had proposed a set that deals with vagueness, imprecision called Fuzzy set from universal set X and [2] give a detailed explanation on this fuzzy set. Fuzzy set explores the characterisation using parameter, linguistic variables etc. Each elements in fuzzy set is represented as membership values from the set X to [0, 1]. It had various applications in image processing, decision making, fuzzy logics and fuzzy inference systems etc. K. Atanassov in 1986, [3] enhances a unique concept called intuitionistic fuzzy sets. It ensures the both membership and non-membership values in intuitionistic fuzzy sets. Mathematical analysis explores the concepts of limits, contuinity, open sets, closed sets, compactness are discussed in Real numbers. After that, various mathematicians convey his ideas and explores his conception to different geometrical space is represented as topology. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6135 Email addresses: thirumaths1996@gmail.com (S. Thirukumaran), gk revathi@yahoo.co.in (G. K. Revathi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 2 of 17 Topological space is other wise expressed as rubber- sheet geometry, it characterizes the shapes and its deformations. Connectedness, compactness and continuity are major three C’s in topological space. Among these, compactness ensures about the open coverings in a give topological spaces. This also leads to a higher dimension topological space called large inductive dimension, short inductive covering dimensions, embeddings, manifolds, functions spaces, and also it leads to a another dimensional concept as algebraic topology. C. L. Chang in 1968, [4], defines fuzzy topological spaces. It enhances a various ideas on structures and spaces also its properties. Later, intuitionistic fuzzy topological space was defined by D. Coker in 1997 [5]. In both fuzzy topological space and intuitionistic fuzzy topological space leads to an concepts for connectedness, compactness etc. B. Hutton in 1975 and 1977 [6, 7] notion on uniformities and normalities in fuzzy topo- logical space. Among all, the main aim in this article is to explores a new essence called (c,d) IF −Q uniform Ir∗ centred structure compactification using (c,d) IF −Q uniform Ir∗ centred structure filters and (c,d) IF −Q uniform Ir∗ centred structue nets. In this concept, various results regarding compactifications and its properties are explored. The relationship between nets and filters in topological spaces is to understand the compact spaces and its characteristics. The (c,d) IF −Q uniform Ir∗ centred structure compacti- fication is motivated by a need to bridge certain gaps in the theory of compactifications of uniform spaces, particularly those involving irregular and quasi-uniform structures. Tra- ditional compactifications, such as Stone–Čech and Samuel compactifications, are largely constructed under assumptions of regularity, symmetry, or completeness. However, many natural and important spaces in both topology and analysis, especially quasi-uniform spaces and structures arising in generalized function theory, lack these properties. The extension of (c,d) IF −Q uniform Ir∗ centred structure compactification more flexible treatment of quasi-uniformities that do not necessarily satisfy classical uniform conditions, enabling a broader class of spaces to be compactified meaningfully. Also the broader field of topology, this research positions itself at the intersection of compactification theory, uniform space theory, and generalized convergence structures. It contributes to the ongo- ing effort to generalize classical results to more flexible, non-standard settings, which are increasingly relevant in modern applications such as rigid motions in theoretical physics, other higher dimension topological spaces, and the study of generalized metric spaces. An filters can associate with nets and vice versa to analyze convergence and compact- ness. Among that the relationship between nets, filters and compactifications are also associated with each other. Filters are used to analyse the compact spaces and construct compactifications. Likewise, the nets are tedious to ensure the convergence in compactified space. 2. Literature Review E. Narmada and et. al [8] had framed a C structure to define an compactification in intuitionistic fuzzy topological spaces. The article, explores an detailed analysis of C structure spaces using TC filters for compactification. In 2014, G. K. Revathi et. al [9] had researched a new approach on Wallman-type compactification via intuitionistic fuzzy S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 3 of 17 rough centred texture spaces. Also in 2015, G. K. Revathi et. al [10] express an novel idea of compactification via semigroup and intuitionistic fuzzy convergence topological spaces. Later on that, in 2015, Ridvan Sahin developed his ideas to soft set. Soft sets deals with paramaters. The author had define a compactification and its properties on soft sets. In 2019, Ceren Sultan ELIMALI et. al [11] defined an FAN-GOTTESMAN compactifications and stone spaces along with properties are discussed. Likewise [12]the topological group of transformations explain about the pointwise convergence topology and admissible group topology with the structure equivariant compactifications. Also [13] introduced the category of stable compactifications and Raney extensions of proximity frames in duality spaces. In this articles, the author defines an comparitive analysis on other type of compactifications. Also they explore his ideas in clopen sets, ultrafilters, non-convergent spaces, etc. An application related to compactifications [14–16] are used in both the theoretical approaches like lie algebra, bounded operators, etc,. and also applied in different fields like physics, robotics, etc. 3. Motivation and Contribution of the Study The major motivation behind the (c,d) IF −Q uniform topological space is an major extension of intuitionsitic fuzzy topological space. A (c,d) IF −Q uniform topological space is a highly generalized mathematical structure that blends concepts from uniform topology, intuitionistic fuzzy sets, and quasi-uniformity. It is designed to model uncer- tainity in topological spaces. This space has led to the numerous concepts in topological space is functors, morphisms etc. Also in (c,d) IF −Q uniform Ir∗ structure space deals with the irreducibility in topological spaces. This leads to Zariski topology, spectral space and Jac-spectral space etc., These are the higher dimension topological space combined with algebraic geometry and commutative algebra. Here, this motivate for novel research ideas which can be incorporated to various domains. Likewise, Centred systems deals only with the collection of open sets in the given topological space. So, here the reserach work incorporates the different spaces in (c,d) IF −Q uniform topological space. Based on the literature survey in Section-2, the authors introduced the novel idea called (c,d) IF −Q uniform Ir topological space. Many researchers depicted and applied compactifications in robotics, physics, lie groups, etc., which leads authors to study (c,d) IF −Q uniform Ir∗ centred structure compactification. To increase the readers interest, the authors intro- duced (c,d) IF −Q uniform Ir∗ centred structure filters and (c,d) IF −Q uniform Ir∗ centred structure nets which supported the study of (c,d) IF −Q uniform Ir∗ centred structure compactification. 4. Proposed Structure of the Paper In this article, an idea of Compactification on (c,d) IF −Q uniform Ir∗ centred struc- ture space is defined. The below flowchart shows the process of compactification is exe- cuted via (c,d) IF −Q uniform Ir∗ centred Filters and (c,d) IF −Q uniform Nets. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 4 of 17 Figure 1: A Study on (c,d) IF −Q uniform Ir∗ centred structure compactification 5. Preliminaries Here IF denotes the intuitionistic fuzzy set on X and throughout the article, and the universe of discourse X is a non-empty set. Definition 1. [3] Let X be a universal set and an IF set A in X is defined as A = {⟨x, µA(x), νA(x)⟩ : x ∈ X} where µA(x) : X → [0, 1] and νA(x) : X → [0, 1] are the mem- bership and non-membership functions respectively for every x ∈ X, with the condition 0 ≤ µA(x) + νA(x) ≤ 1. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 5 of 17 Definition 2. [5] An IF topology τ on X is a collection of intuitionistic fuzzy sets, the following assertion are to be hold: (i) 0∼,1∼ ∈ τ , (ii) Each Ai ∈ τ , then ∩n i=1Ai ∈ τ . (iii) For all Ai ∈ τ then ∪i ∈ JA ∈ τ . The pair (X, τ) is referred to as IF topological space on X Definition 3. [17] Suppose ΞX represents the collection of IF mappings, g : IF (X) → IF (X) then the following are: (i) g(0∼) =0∼ (ii) A ⊆ g(A), ∀A ∈ IF(X) (iii) g(∪Ai)= ∪g(Ai), ∀Ai ∈ IF (X), i ∈ J For g ∈ ΞX , the function g−1(A)= ∩{B : g(B̄) ⊆ Ā} ∈ ΞX , given for all A ∈ IF (X), g ⊓ g ′ (A)=∩{g(A1) ∪ g ′ (A2) : A1 ∪A2 = A}, (g ◦ g′) (A)=g(g ′ (A)). Definition 4. [17] Let υ: ΞX → I × I be an IF mapping. Then υ is characterized as an IF quasi uniformity on X, if it fulfills the circumstances: (i) υ(g1 ⊓ g2) ⊇ υ(g1) ∩ υ(g2) for g1, g2 ∈ ΞX (ii) g ∈ ΞX we have ∪{υ(g1) : g1 ◦ g1 ⊆ g} ⊇ υ(g) (iii) g1 ⊇ g then υ(g1) ⊇ υ(g) (iv) g ∈ ΞX then υ(f)=1∼. A pair (X, υ) is claimed as IF −Q uniform space. Definition 5. [17] A pair (X, υ) be an IF −Q uniform space. Let c ∈ (0, 1] =I0 and d ∈ [0, 1) =I1 with c + d ≤ 1 and A ∈ IF (X). (c,d)IFQIυ(A) = ∪{B : f(B) ⊆ A, some of f ∈ Ξ(X) and υ(f) > (c,d)} Definition 6. [17] Consider (X, υ) be an IF −Q uniform space The mapping τυ is defined by: IF (X)→ I × I is defined by τυ(A) = ∪{(c,d) : (c,d) IFQIυ(A)=A, c ∈ I0, d ∈ I1 with c + d ≤ 1}.Hence, a pair (X, τυ) is called IF −Q uniform topological space. The elements of (X, τυ) is called (c,d) IF −Q uniform open sets and its complement is (c,d) IF −Q uniform closed sets. Example 1. Assume X = {w,y}. A = 〈 x, ( w 0.3 , y 0.5 ) , ( w 0.4 , y 0.1 )〉 , B = 〈 x, ( w 0.1 , y 0.2 ) , ( w 0.5 , y 0.6 )〉 and C = 〈 x, ( w 0.3 , y 0.2 ) , ( w 0.4 , y 0.3 )〉 be any three IF sets on X and let E is a non-void IF set. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 6 of 17 Let ΞX: IF (X) → IF (X) be an IF mapping. Let g1, g2, g3 and g4 ∈ ΞX be defined as: g1(E) = { 0∼, ifE = 0∼ 1∼, otherwise g2(E) =  0∼, ifE = 0∼ A, ifE ⊆ A 1∼, otherwise. g3(E) =  0∼, ifE = 0∼ B, ifE ⊆ B 1∼, otherwise. g4(E) =  0∼, ifE = 0∼ C, ifE ⊆ C 1∼, otherwise. υ(g) =  (1, 0), ifg = g1 (2/7, 5/8), ifg = g2 (3/6, 1/5), ifg = g3 (5/9, 9/11), ifg = g4 (4/7, 1/9), ifg = g2 ⊓ g3 (5/7, 8/9), ifg = g2 ⊓ g4 (5/9, 6/7), ifg = g3 ⊓ g4 (0, 1), otherwise Clearly, (X, υ) is an IF −Q uniform space, for c=0.02 and d=0.05. define intuitionistic fuzzy mapping, τυ: IF (X) → I × I as τυ(E) =  (0, 1), ifE = 0∼ (4/5, 2/7), ifE = A (2/5, 3/7), ifE = B (1/8, 2/7), ifE = C (1, 0), otherwise Hence τυ={A,B,C, 0∼, 1∼}. Clearly (X, τυ) is an IF −Q uniform topological space. Definition 7. [17] A pair (X, τυ) be an IF −Q uniform topological space and A be an IF set. The IF −Q uniform interior of A is then described as (c,d)IFQintυ(A)=∪{B : B ⊆ A and B is a (c,d) IF −Q uniform open set where c ∈ I0, d ∈ I1, and c+d ≤ 1}. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 7 of 17 Definition 8. [17] Let (X, τυ) be an IF −Q uniform topological space and A be a IF set. Then the IF −Q uniform closure of A is expressed as (c,d)IFQclυ(A)=∩{B : B ⊇ A and B is an (c,d) IF −Q uniform closed set where c ∈ I0, d ∈ I1 with c+d ≤ 1} 6. Compactification in (c,d) IF −Q Uniform Ir∗-structure space Throughout the article the word ”bounded” is represented as ”bdd”. 6.1. Irreducibility and (c,d) IF −Q Uniform Ir∗ Centred Structure space in IF −Q Uniform Topological space Definition 9. Let (X, τυ) represents an IF −Q uniform topological space. A,B and C ∈ IF (X). An IF set A is said to be irreducible iff A ⊆ B ∪ C, such that A ⊆ B or A ⊆ C with B ̸= 1∼ and C ̸= 1∼ Definition 10. A pair (X, τυ) be IF −Q uniform topological space and A,B, C be any (c,d) IF −Q uniform open sets is said be to irreducible, iff A ⊆ B ∪ C such that A ⊆ B or A ⊆ C. Then A is said to be (c,d) IF −Q uniform irreducible open set. The complement of (c,d) IF −Q uniform irreducible open sets is said to be (c,d) IF −Q uniform irreducible closed set. Example 2. Let X = {g,h} be a non empty set, A = 〈 x, ( g 0.1 , h 0.2 ) , ( g 0.3 , h 0.5 )〉 B = 〈 x, ( g 0.2 , h 0.3 ) , ( g 0.2 , h 0.2 )〉 , C = 〈 x, ( g 0.4 , h 0.4 ) , ( h 0.1 , h 0.1 )〉 and D = 〈 x, ( g 0.8 , h 0.1 ) , ( g 0.03 , h 0.01 )〉 be any four IF sets on X and let E is a non-empty IF set on X. Consider the IF mapping, ΞX: IF (X) → IF (X) g1, g2, g3, g4 and g5 ∈ ΞX be defined as: g1(E) = { 0∼, ifE = 0∼ 1∼, otherwise. g2(E) =  0∼, ifE = 0∼ A, ifE ⊆ A 1∼, otherwise. g3(E) =  0∼, ifE = 0∼ B, ifE ⊆ B 1∼, otherwise. g4(E) =  0∼, ifE = 0∼ C, ifE ⊆ C 1∼, otherwise. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 8 of 17 g5(E) =  0∼, ifE = 0∼ D, ifE ⊆ D 1∼, otherwise. υ(g) =  (1, 0), ifg = g1 (2/7, 5/8), ifg = g2 (3/6, 1/5), ifg = g3 (3/7, 1/6), ifg = g4 (3/8, 1/8), ifg = g5 (4/9, 1/9), ifg = g2 ⊓ g3 (5/7, 5/9), ifg = g2 ⊓ g4 (6/7, 6/9), ifg = g2 ⊓ g5 (3/7, 8/9), ifg = g3 ⊓ g4 (4/9, 1/10), ifg = g3 ⊓ g5 (1/6, 4/7), ifg = g4 ⊓ g5 (0, 1), otherwise Clearly, (X, υ) is an IF −Q uniform space, for c=0.08 and d=0.03. define an IF mapping, τυ: IF (X) → I × I as τυ(E) =  (0, 1), ifE = 0∼ (2/5, 3/7), ifE = A (1/8, 2/7), ifE = B (1/3, 3/8), ifE = C (2/5, 3/5), ifE = D (1, 0), otherwise Then τυ={A,B,C,D, 0∼, 1∼} be an IF −Q uniform topological space. The elements of τυ are called (c,d) IF −Q uniform open sets and the complements are (c,d) IF −Q uniform closed sets. Clearly (X, τυ) is an IF −Q uniform topological space. Let A,B and C are (c,d) IF −Q uniform irreducible open sets. From the Definition 6.2, the collection Ir= {A,B,C, 0∼} are (c,d) IF −Q uniform irreducible open sets. Remark 1. From the above Example 6.3, Clearly Ir∗= Ir∪{1∼}. Then Ir∗= {A,B,C, 0∼}∪ 1∼ is said to be (c,d) IF −Q uniform Ir∗ irreducible open sets. Definition 11. Let (X, τυ) be an IF −Q uniform topological space on X. Then (X, τυ) is said to be (c,d) IF −Q uniform Ir∗ structure space, then the corresponding requirements are to be fulfilled: (i) 0∼, 1∼ ∈ Ir∗. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 9 of 17 (ii) If {Ai : i ∈ I, ∀Ai ∈ Ir∗} where ∪i∈I Ai ∈ Ir∗. (iii) If A,B ∈ Ir∗, then A ∩ B ∈ Ir∗. Every member of (X, τυ) is said to be (c,d) IF −Q uniform Ir∗ structure space. The elements of (c,d) IF −Q uniform Ir∗ structure space is (c,d) IF −Q uniform Ir∗ struc- ture open sets. The complements of (c,d) IF −Q uniform Ir∗ structure open set is (c,d) IF −Q uniform Ir∗ structure closed sets. Definition 12. Let (X,τυ) be IF −Q uniform topological space. Then (X,τυ) is said to be (c,d) IF −Q uniform Ir∗ structure hausdorff space, iff for every x1, x2 ∈ X and x1 ̸= x2 implies that there exists G1= ⟨x, µG1 , νG1⟩, G2= ⟨x, µG2 , νG2⟩ ∈ τυ with µG1(x1)= 1∼, νG1(x1)= 0∼, µG2(x2)= 1∼, νG2(x2)= 0∼ and G1 ∩G2= 0∼ Definition 13. Let (X, τυ) is said to be (c,d) IF −Q uniform Ir∗ structure hausdorff space. The collection P = {Ai}i∈δ of all (c,d) IF −Q uniform Ir∗ structure open sets of (X, τυ) referred to as (c,d) IF −Q uniform Ir∗ centred structure system, if for any finite collection of elements in IF −Q uniform Ir∗ structure space such that ∩n i=1Ai ̸= 0∼. Definition 14. Consider, the sets PX= {Pi : i ∈ δ} where P , is are (c,d) IF −Q uniform Ir∗ centred structure systems in (X, τυ) which are also called as (c,d) IF −Q uniform Ir∗ centred structure points. Then the family τP is said to be an (c,d) IF −Q uniform Ir∗ centred structure, if it satisfies the following conditions: (i) ∅, PX ∈ τP (ii) ∪i∈J τP is in τP .(arbitrary union) (iii) ∩n i=1 τP is in τP .(finite intersection) The pair (PX , τP ) is called (c,d) IF −Q uniform Ir∗ centred structure space. Each members of (PX , τP ) are called (c,d) IF −Q uniform Ir∗ centred structure open set. The complement of an (c,d) IF −Q Ir∗ centred structure uniform open set is (c,d) IF −Q uniform Ir∗ centred structure closed set. Definition 15. Let (PX ,τP ) be (c,d) IF −Q uniform Ir∗ centred structure space and A ⊆ PX . Then (c,d) IF −Q uniform Ir∗ centred structure closure and (c,d) IF −Q uniform Ir∗ centred structure interior of A is defined as Ir∗ Cp Cl(A)=∩{B : B is an (c,d) IF −Q uniform Ir∗ centred structure closed set and A ⊆ B} Ir∗ Cp Int(A)=∩{B : B is an (c,d) IF −Q uniform Ir∗ centred structure closed set and A ⊇ B} Definition 16. Let (PX ,τP ) be the (c,d) IF −Q uniform Ir∗ centred structure space. A collection {Ai : i ∈ δ} of (c,d) IF −Q uniform Ir∗ centred structure open sets in a (c,d) IF −Q uniform Ir∗ centred structure space (PX , τP ) is called a (c,d) IF −Q uniform Ir∗ centred structure open cover of (c,d) IF −Q uniform Ir∗ centred subset B of PX , if B ⊆ ∪ {Ai : i ∈ δ}. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 10 of 17 Definition 17. Let (PX ,τP ) be the (c,d) IF −Q uniform Ir∗ centred structure space is said to be a (c,d) IF −Q uniform Ir∗ centred structure compact, if for every (c,d) IF −Q uniform Ir∗ open cover of PX possess a finite subcover. Definition 18. A pairs (PX , τP ) and (PY , τ ∗ P ) be any two (c,d) IF −Q uniform Ir∗ centred structure space. Then f: (PX , τP ) → (PY , τ ∗ P ) is an (c,d) IF −Q uniform Ir∗ centred structure continuous function, if f−1(V ) is an (c,d) IF −Q uniform Ir∗ centred structure open set in (PX , τP ) for each (c,d) IF −Q uniform Ir∗centred open set V in (PY , τ ∗ P ). Definition 19. Let (PX , τP ) be a (c,d) IF −Q uniform Ir∗ centred structure space and let P ∈ PX . Then a (c,d) IF −Q uniform Ir∗ centred structure subset N ⊆ PX is said to be (c,d) IF −Q uniform Ir∗ structure neighborhood, if there exists a (c,d) IF −Q uniform Ir∗ structure open sets G such that P ∈ G ⊆ N . Definition 20. Let (PX , τP ) be a (c,d) IF −Q uniform Ir∗ centred structure space. A subset A of PX is said to be a (c,d) IF −Q uniform Ir∗centred structure dense, if Ir∗CP cl(A)=PX . Definition 21. A pair (PX , τP ) be a (c,d) IF −Q uniform Ir∗ centred structure space. Then a non-empty family F of subsets of PX is called (c,d) IF −Q uniform Ir∗ centred structure filter on PX , iff it satisfies the following conditions: (i) ∅ ∈ F (ii) Assume F ∈ F and F ⊆ H, then H ∈ F. (iii) Consider F1, F2 ∈ F, then F1 ∩ F2 ∈ F Definition 22. Consider (c,d) IF −Q uniform Ir∗ centred structure net in an (c,d) IF −Q uniform Ir∗ centred structure space (PX , τP ) is a function from a directed set ∆ to PX . It is denoted as {Pζ}ζ∈ς 6.2. Nets, Filters and Convergence in (c,d) IF −Q Uniform Ir∗ Centred Structure space Notation: Here, throughout this article the notation ⊘ is used for eventually and ⊖ is used for frequently Definition 23. Let { Pζ } ζ∈ς be a (c,d) IF −Q uniform Ir∗ centred structure net in an (c,d) IF −Q uniform Ir∗ centred structure space PX and let G be a (c,d) IF −Q uniform Ir∗ centred structure subset of PX . Then the (c,d) IF −Q uniform Ir∗ centred structure net is expressed as (i) in G iff {Pζ} ∈ G, ∀ζ ∈ ς. (ii) ⊘ ∈ G iff there is an existence of ϑ ∈ ς∀α ∈ ς α ≥ ϑ, {Pζ} ∈ G S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 11 of 17 (iii) ⊖ ∈ G, iff for all ϑ ∈ ς, there is an existence of ζ ∈ ς, ζ ≥ ϑ and {Pζ} ∈ G Definition 24. Let {Pζ} is a (c,d) IF −Q uniform Ir∗ centred structure net in the (c,d) IF −Q uniform Ir∗ centred structure PX and P is a (c,d) IF −Q uniform Ir∗ centred structure element of PX . An (c,d) IF −Q uniform Ir∗ centred structure net converges towards P iff for every (c,d) IF −Q uniform Ir∗ centred structure neighborhood U of P , such that {Pζ} ⊘ in U . Definition 25. A pair (c,d) IF −Q uniform Ir∗ centred structure element P1 of PX is said to be a (c,d) IF −Q uniform Ir∗ centred structure accumulation point or cluster point of a (c,d) IF −Q uniform Ir∗ centred structure net iff for every (c,d) IF −Q uniform Ir∗ centred structure neighborhood U of P1, so that (c,d) IF −Q uniform Ir∗ centred structure net is ⊖ in U . Definition 26. Let (c,d) IF −Q uniform Ir∗ centred structure net {P} in a set PX is said to be (c,d) IF −Q uniform Ir∗ centred structure universal or (c,d) IF −Q uniform Ir∗ centred structure ultranet, if for every (c,d) IF −Q uniform Ir∗ centred structure subset A of PX , either {P} is ⊘ in A or {P} is ⊘ in PX −A. Definition 27. A collection ϕ represents (c,d) IF −Q uniform Ir∗ centred structure continuous function on an (c,d) IF −Q uniform Ir∗ centred structure space PX . A (c,d) IF −Q uniform Ir∗ centred structure net {Pi} in PX will be called as (c,d) IF −Q uniform Ir∗ centred structure ϕ net, converges to each f in the mapping ϕ. Definition 28. A mapping ϕ contains a collection of bdd real valued continuous function on PX . Here, PX is a (c,d) IF −Q uniform Ir∗ centred structure compact set, for every ϕ net has an (c,d) IF −Q uniform Ir∗ centred structure cluster point in PX . Definition 29. Let PX be an (c,d) IF −Q uniform Ir∗ centred structure space, C∗(PX)= {fζ : ζ ∈ ς} be the collection of all bdd real-valued continuous function on PX . Regarding a C∗(PX) net {Pi} and F{Pi}= {U : U is a (c,d) IF −Q uniform Ir∗ centred structure open set in PX and {Pi} is ⊘ in U}. By knowing that, F{Pi} is an (c,d) IF −Q uniform Ir∗ centred structure open filter and some fζ ∈ C∗(PX), any ϵ > 0, (fζ) −1((rζ−ε, rζ+ε)) ∈ F{Pi} where rζ= lim {fζ(Pi)} F{Pi} is the (c,d) IF −Q uniform Ir∗ centred structure open filter in PX induced by {Pi}. Definition 30. Let F is a (c,d) IF −Q uniform Ir∗ centred strcture filter on PX . Let ςF= {(P, F ) : P ∈ F ∈ F }. So ςF is expressed as the relation (P1, F1) ≤ (P2, F2) iff F2 ⊂ F1 and the function M : ςF → PX expressed as M(P, F1) = P is an (c,d) IF −Q uniform Ir∗ centred structure Net in PX . It is called as (c,d) IF −Q uniform Ir∗ centred net based on F . Definition 31. The pair (c,d) be an IF −Q uniform Ir∗ centred structure filter FP converges to P in PX , if the (c,d) IF −Q uniform Ir∗ centred structure net converges to P with respect to FP . S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 12 of 17 Definition 32. Let P be a (c,d) IF −Q uniform Ir∗ centred structure open filter on PX and {Pi} be an (c,d) IF −Q uniform Ir∗ centred net related to P , and I= {U : U is a (c,d) IF −Q uniform Ir∗ centred structure open in PX also {Pi} is ⊘ in U}. Then I = P Remark 2. Every C∗(PX) net {Pi} in PX and also {wPi k } be the (c,d) IF −Q uniform Ir∗ centred Net based on the (c,d) IF −Q uniform Ir∗ centred structure open filter F{Pi} induced by {Pi}. (i) {wPi k } is distinctly established as F{Pi} and F{Pi}= F{Pj} iff {wPi k }= {wPj k }. (ii) F{Pi}=F{wPi k } = {O : O is (c,d) IF −Q uniform Ir∗ centred structure open set in PX and {wPi k } is eventually in O } (iii) {wPi k } is a C∗(PX) net along with lim {fζ(wPi k )}= lim {fζ(Pi)} ∀ fζ in C∗(PX). (iv) requirements are to be hold: (a) {wPi k } convergent to P (b) Here, converged to P with respect to {Pi}. (c) F{Pi} converges to P . Let YP= {{wPi k }∗ : {Pi} is an C∗(PX) net does not converge to PX , {wPi k } is (c,d) IF −Q uniform Ir∗ centred structure Net based on (c,d) IF −Q uniform Ir∗ centred structure filter F{Pi}}, P ∗ X= PX ∪ YP , the disjoint union of PX and YP . For each (c,d) IF −Q uniform Ir∗ centred structure open set U ⊂ PX define U∗ ⊆ P ∗ X and the set U∗= U ∪ {{wPi k }∗ : {wPi k }∗} ∈ YP and {wPi k } is ⊘ in U}. It is apparent that if U ⊂ V , then U∗ ⊂ V ∗. Proposition 1. Let U and V be any two (c,d) IF −Q uniform Ir∗ centred structure open sets in PX , then (U ∩ V )∗= U∗ ∩V ∗. Proof. Let P2 ∈ (U ∩ V )∗ ∩YP , then P2= {wPi k }∗ and {wPi k } is ⊘ in U ∩ V . This indicates that {wPi k } is ⊘ ∈ U ,V . Here, {wPi k }∗ ∈ U∗ ∩ V ∗. If P2 ∈ (U∗ ∩ V ∗) ∩ YP , then P2= {wPi k }∗ and {wPi k } is ⊘ is both in U and V . So {wPi k } is ⊘ ∈ U ∩ V . Then P2 ∈ (U ∩ V )∗ Proposition 2. Assume B= {U∗: U be an (c,d) IF −Q uniform Ir∗ centred structure open set in PX }. Then B is an (c,d) IF −Q uniform Ir∗ centred structure base for an (c,d) IF −Q uniform Ir∗ centred structure on P ∗ X , if (i) P ∗ X = {U∗ : U∗ ∈ B} (ii) Each U∗, V ∗ ∈ B with P2 ∈ U∗ ∩ V ∗ some of the W ∗= (U∗ ∩ V ∗) ∈ B, P2 ∈ W ∗ ⊂ U∗ ∩ V ∗ Proof. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 13 of 17 (i) P ∗ X = {U∗ : U∗ ∈ B}, let P2 ∈ YP , then P2= {wPi k }∗. For any fζ ∈ C∗(PX), let rζ= lim {fζ(wPi k ), then {wPi k } is ⊘ in f−1 α ((rζ − ε, rζ + ε)), for any ε > 0, that is {wPi k }∗ is in f−1 ζ ((rζ −ε, rζ +ε)) for all υ > 0, accordingly YP ⊂ ∪{U∗ : U∗ ∈ B} consequently P ∗ X ⊂ ∪{U∗ : U∗ ∈ B}. For {U∗ : U∗ ∈ B} ⊂ P ∗ X is explained. (ii) if P2 ∈ U∗ ∩ V ∗, for any U∗ and V ∗ in B, since (U ∩ V )∗ is in B and (U ∩ V )∗= U∗ ∩ V ∗, thus P2 ∈ (U ∩ V )∗ ⊂ U∗ ∩ V ∗. Remark 3. Consider the P ∗ X with (c,d) IF −Q uniform Ir∗ centred structure induced by the (c,d) IF −Q uniform Ir∗ centred structure base B. For all fζ in C∗(PX), defined f∗ζ : P ∗ X → R define that f∗ζ(P1)= fζ(P1) if P1 ∈ PX , f∗ζ({w Pi k }∗)= lim {fζ(w Pi k )} for any {wPi k }∗ in YP and it is stated as f∗ζ is clearly specified that it is bdd real valued function on P ∗ X. Proposition 3. For Any fζ in C∗(PX), f∗ζ is a bounded real valued continuous function on P ∗ X . Proof. It is enough to proof the continuity, f∗ζ at any P3 in P ∗ X , let tζ=f∗ ζ (P3). It will be shown that for any ε > 0, there is an (c,d) IF −Q uniform Ir∗ centred structure open set U∗ ∈ B such that P3 ∈ U∗ ⊂ (f∗ ζ ) −1((tζ − ε, tζ + ε)). Let U=f−1 ζ ((tζ − ε/2, tζ + ε/2)). If P3 ∈ PX , since fζ(P3) = f∗ ζ (P3) = tζ , thus P3 ∈ f−1 ζ ((tζ − ε/2, tζ + ε/2)) ⊂ (f−1 ζ ((tζ − ε/2, tζ + ε/2)))∗. If P3 ∈ YP , then P3= {wPi k }∗. Since tζ = f∗ ζ (P3)= lim {fζ(wPi k ), so {wPi k } is ⊘ in f−1 ζ ((tζ − ε/2, tζ + ε/2)). that is P3 = {wPi k }∗ ∈ (f−1 ζ ((tζ − ε/2, tζ + ε/2)))∗. Finally show that, (f−1 ζ ((tζ − ε/2, tζ + ε/2)))∗ ⊂ (f∗ ζ ) −1((tζ − ε, tζ + ε)). If P1 is in PX∩ (f−1 ζ ((tζ − ε/2, tζ + ε/2)))∗ then P1 ∈ f−1 ζ ((tζ − ε/2, tζ + ε/2)) that is f∗ ζ (P1)= fζ(P1) ∈ (tζ − ε, tζ + ε). So, P1 ∈ (f∗ ζ ) −1((tζ − ε, tζ + ε)). If P2 ∈ (f−1 ζ ((tζ − ε/2, tζ + ε/2)))∗ ∩YP , then P2= {wPi k }∗ and {wPi k } is ⊘ in (fζ) −1((tζ −ε/2, tζ +ε/2)) thus f∗ ζ (P2)= lim {fζ(wPi k )} ∈ [tζ − ε/2, tζ + ε/2] ⊂ (tζ − ε, tζ + ε) that is P2 ∈ (f∗ ζ ) −1((tζ − ε, tζ + ε)). Proposition 4. Let K : PX → P ∗ X be defined by K(P1) = P1, then K is an (c,d) IF −Q uniform Ir∗ centred structure continuous mapping from PX into P ∗ X . Proof. For any (c,d) IF −Q uniform Ir∗ centred structure open set U∗ ∈B,K−1(U∗) = U is a (c,d) IF −Q uniform Ir∗ centred structure open set in PX, so K is an (c,d) IF −Q uniform Ir∗ centred structure continuous mapping on PX. Proposition 5. For any P2 in P ∗ X − PX with P2 = {wPi k }∗, {k(wPi k )} converges to P2= {(wPi k )}∗. Proof. Let U∗ be any (c,d) IF −Q uniform Ir∗ centred structure open set in B containing P2 then {(wPi k } and ⊘ ∈ U in PX. This implies that {k(wPi k )} is ⊘ in U∗, thus {K(wPi k )} it is converged to P2={wPi k }∗. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 14 of 17 Proposition 6. K(PX) is an (c,d) IF −Q uniform Ir∗ centred structure dense in P ∗ X Proof. For any P2 in P ∗ X −PX. P2 = {wPi k }∗. By the above Proposition 5, implies that {K(wPi i )} converges to P2={wPi k }∗. Thus Ir∗ Cp Cl(K(PX))=P ∗ X. Remark 4. Here, C={f∗ ζ : fζ ∈ δ} represent {f∗ ζ : fζ ∈ C∗(PX)}. Each C net Pi in P ∗ X and E = {O : O is an (c,d) IF −Q uniform Ir∗ centred structure open in P ∗ X and {Pi} is ⊘ in O}. L = {U : U is an (c,d) IF −Q uniform Ir∗ centred structure open in PX and U∗ ∈ E}. Proposition 7. For a C net {Pi}, in P ∗ X. Let rζ= lim {f∗ ζ (Pi)}. All f∗ ζ ∈ C. Then for arbitrary ε > 0, (f∗ ζ ) −1((rζ − ε, rζ + ε)) ⊂ (f−1 ζ ((rζ − ε, rζ + ε)))∗. Proof. Consider, P3 ∈ (f∗ ζ ) −1((rζ − ε, rζ + ε)), then f∗ ζ (P3) ∈ (rζ − ε, rζ + ε). If P3 =K(P1) = P1,∀ P ∈ PX . Given that, fζ(P1)= f∗ ζ (P3), so P1 is in (fζ) −1((rζ − ε, rζ + ε))∗. If P3= {wPi k }∗ in YP , then lim {fζ(wPi k )}= f∗ ζ (P3) ∈ (rζ − ε, rζ + ε). This implies that {wPi k } is ⊘ in (fζ) −1 ((rζ − ε, rζ + ε)) thus {wPi k } is in (fζ) −1(rζ − ε, rζ + ε)∗. Corollary 1. For a C net {Pi} in P ∗ X . Let rα=lim {f∗ ζ (Pi)} for every f∗ ζ ∈ C. Then for arbitrary ε > 0, (f∗ ζ ) −1((rζ − ε, rζ + ε)) ∈ E and (fζ) −1((rζ − ε, rζ + ε)) ∈ L. Proof. Here, (f∗ ζ ) −1 ((rζ − ε, rζ + ε)) ∈ E. By the above Proposition 7, {Pi} is ⊘ in (fζ) −1((rζ − ε, rζ + ε))∗, thus (fζ) −1((rζ − ε, rζ + ε)) ∈ L - Proposition 8. E and L are (c,d) IF −Q uniform Ir∗ centred structure open filter on P ∗ X and PX respectively. Proof. Building on the proof of Proposition 1 and Corollary 1 , it is evident that E is an (c,d) IF −Q uniform Ir∗ centred structure filter on P ∗ X. By Corollary 1 L ̸= ∅. If U , V are (c,d) IF −Q uniform Ir∗ centred strcuture open sets in L, then U∗ and V ∗ ∈ E. Since (U ∩ V )∗= U∗ ∩ V ∗ and U∗ ∩ V ∗ ∈ E thus U ∩ V ∈ L. If W is an (c,d) IF −Q uniform Ir∗ centred structure open set both W ⊃ O and W ∗ ⊃ Ø∗. Hence it is implies that W ∗ ∈ E and W ∈ L. Proposition 9. The C net {Pi} converges with respect to P ∗ X . Proof. Let {wk} be the (c,d) IF −Q uniform Ir∗ basis on centred structure net L. Since for any α ∈ δ and ε > 0, (f−1 ζ )((rζ − ε, rζ + ε)) ∈ L, where rζ=lim {f∗ ζ Pi}. So, fζ(wk) converges to rζ ∀ ζ ∈ δ. i.e {wk} is a C∗(PX) net. Since (c,d) IF −Q uniform Ir∗ centred structure open filter F{wk} formed by the C∗(PX) net {wk} is exactly in L. So, if {wPk k } is the (c,d) IF −Q uniform Ir∗ centred structure net according to F{wk}, then {wk}= {wPk k }. Case 1: If {wk} converges to an (c,d) IF −Q uniform Ir∗ centred structure point P in PX. Let U ∗ be an (c,d) IF −Q uniform Ir∗ centred structure open set B contains K(P ), then P ∈ U .Here, U is an (c,d) IF −Q uniform Ir∗ centred structure open set in PX. Since {wk} converges to P , by considering Definition 31, U in L and therefore U∗ is in E. It states that {Pi} converged to K(P ) in P ∗ X. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 15 of 17 Case 2: If {wk} diverges to {PX} then {w∗ k}= {wPk k }∗ is in YP . For all U ∗ is in B containing {wPk k } is ⊘ in U in PX then by Definition 32 implies that U in L and therefore U∗ is in E . Thus {Pi} converges to {wPk k }∗= {wk}∗ ∈ P ∗ X. Proposition 10. (P ∗ X,K) is an (c,d) IF −Q uniform Ir∗ centred structure compactifi- cation of PX. Proof. A family of C is said to be bdd real-valued continuous functions on P∗ X and for all C net {Pi} converges to P ∗ X. From the Definition 28, P ∗ X is an (c,d) IF −Q uniform Ir∗ centred structure compact space. Here, the Proposition 6, implies that (P ∗ X,K) is an (c,d) IF −Q uniform Ir∗ centred structure compactification of PX. Proposition 11. Let C(P ∗ X) be the collection of all real valued continuous functions on P ∗ X . Then C(P ∗ X)= C= {f∗ ζ : fζ ∈ C∗(PX)} Proof. Let g ∈ C(P ∗ X). Since P∗ X is an (c,d) IF −Q uniform Ir∗ centred structure compact, so g ◦K ∈ C(P ∗ X). By Proposition 5 and 6 along with based on the continuity of g, it follows that (g◦K)∗({wPi k }) = lim {(g◦K)(wPi i )}= lim {g(K(wPi k ))}=g(lim {K(wPi K )}) =g({wPi k }∗) ∀ {wPi k } ∈ YP and (g ◦K)∗ (K(P ))= (g ◦K)∗ (P) = g(K(P )) ∀ P ∈ PX . It is stated that, C(P ∗ X) ⊂ C= {f∗ ζ : fζ ∈ C∗(PX)}. 7. Results and Discussions The (c,d) IF −Q uniform Ir∗ Centred structure compactification is an unique ap- proach of compactification methods compared to others. Here, it was studied with the irreducibility, centred systems, compact spaces, Nets and filters etc., Also with convergence using boundedness and continous functions. While comparing to other compactifications as mentioned in Section-2 Literature survey. Fan-Gottesmann compactification deals with clopen open sets with filters and nets alongs with convergence not using the boundedness. Also, some of the compactifications make a way to conceptual ideas of framing from non- compact spaces to compact spaces and others. Apart from all these, (c,d) IF −Q uniform Ir∗ Centred Structure compactification is unique one which is being centred system and irreducible sets. 8. Conclusion This research work will lead to a fine understanding about (c,d) IF −Q uniform Ir∗ centred structure space and its properties. It enhances the theoretical approaches to var- ious contexts in compactifications. The inter-relations between (c,d) IF −Q uniform Ir∗ centred structure nets and (c,d) IF −Q uniform Ir∗ centred structure filters pro- vide a robust theoretical framework for understanding (c,d) IF −Q uniform Ir∗ centred structure compactness in various types of spaces. Likewise, (c,d) IF −Q uniform Ir∗ centred structure nets provides a way to generalize sequences and convergence in (c,d) S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 16 of 17 IF −Q uniform topological space while (c,d) IF −Q uniform Ir∗ centred structure fil- ters discussed a framework for convergence and compactness. It will lead to applications of compactifications on various fields. Future framework of (c,d) IF −Q uniform Ir∗ cen- tred structure compactification can be explored into category theory, fixed point theory along with metric spaces. Acknowledgements The authors are highly thankful to the referees and editors for their valuable comments and suggestions to our article. References [1] Lotfi Asker Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965. [2] Hans-Jürgen Zimmermann. Fuzzy set theory—and its applications. Springer Science & Business Media, 2011. [3] Krassimir T Atanassov. On intuitionistic fuzzy sets theory, volume 283. Springer, 2012. [4] Chin-Liang Chang. Fuzzy topological spaces. Journal of mathematical Analysis and Applications, 24(1):182–190, 1968. [5] Doǧan Çoker. An introduction to intuitionistic fuzzy topological spaces. Fuzzy sets and systems, 88(1):81–89, 1997. [6] Bruce Hutton. Normality in fuzzy topological spaces. Journal of Mathematical Anal- ysis and Applications, 50(1):74–79, 1975. [7] Bruce Hutton. Uniformities on fuzzy topological spaces. Journal of Mathematical Analysis and Applications, 58(3):559–571, 1977. [8] R Narmada Devi, E Roja, and MKUma. A new view on intuitionistic fuzzy c structure compactification. Annals of Fuzzy Mathematics and Informatics, 5(3):571–582, 2013. [9] GK Revathi, R Narmada Devi, and E Roja. Intuitionistic fuzzy rough centred tex- ture wallman type compactification. Annals of Fuzzy Mathematics and Informatics, 7(4):699–714, 2014. [10] GK Revathi, E Roja, and MK Uma. Semigroup compactification in an intuitionistic fuzzy convergence topological space. Italian Journal of Pure and Applied Mathemat- ics, (35):9–22, 2015. [11] Ceren Sultan Elmali and Tamer Ugur. Fan-gottesman compactifications and stone space. Sigma, 10(2):143–147, 2019. [12] Boris Vladimirovich Sorin. Compactifications of homeomorphism groups of linearly ordered compacta. Mathematical Notes, 112(1):126–141, 2022. [13] G Bezhanishvili and J Harding. Duality theory for the category of stable compactifi- cations. In Topology Proc, volume 61, pages 1–13, 2023. [14] J Flachsmeyer and F Terpe. Some applications of the theory of compactifications of topological spaces and measure theory. Russian Mathematical Surveys, 32(5):133, 1977. S. Thirukumaran, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6135 17 of 17 [15] Klaus G Witz. Applications of a compactification for bounded operator semigroups. Illinois Journal of Mathematics, 8(4):685–696, 1964. [16] Nestor Djintelbe and Michel Coste. Compactification of the group of rigid motions and applications to robotics. Journal of Pure and Applied Algebra, 225(7):106604, 2021. [17] GK Revathi, E Roja, and MK Uma. A new approach to intuitionistic fuzzy quasi uniform regular gδ compactness. International Journal of Mathematics Sciences and Applications, 2, 2012.