EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 638-662 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fuzzy SSPO-separation Axioms and Fuzzy α-SSPO Compactness Shkumbin Makolli1,∗, Biljana Krsteska2 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Prishtina, Prishtina, Republic of Kosovo 2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Saint Cyril and Methodius, Skopje, Republic of North Macedonia Abstract. In this paper, we introduce the concept of new separation axioms named fuzzy SSPO- separation axioms by using the fuzzy strong semi preo-pen sets and we also introduce and inves- tigate properties of α-SSPO compactness. We define and investigate the relation between fuzzy separation axioms, fuzzy pre-separation axioms, and different forms of fuzzy continuous mappings. We also investigate the existence of a countable base of fuzzy strong semi pre-open sets, we define the concept of SSPO separability, the concept of α−SSPO Lindelof sets and examine their proper- ties. With the concepts of fuzzy strong semi pre-continuity, SSPO-irresolute continuous mappings, and other forms of fuzzy continuity, we investigate the new concept of fuzzy compactness and its properties in regard to the mentioned mappings. 2020 Mathematics Subject Classifications: 54A40, 03E72 Key Words and Phrases: Fuzzy separation axioms, Fuzzy compactness, Fuzzy topological space, Fuzzy strongly semi pre-open set, Fuzzy continuity, Fuzzy SSPO-irresolute continuous mapping, Fuzzy SSPO-irresolute open (closed) mapping, Fuzzy SSPO homeomorphism 1. Introduction Separation axioms were introduced to fuzzy topological spaces in [9], [10], [11],[24],[32], and in some more recent works [25], [28]. They were extensions of separation axioms introduced in general Topology. The separation axioms are more restrictive in the fuzzy topologies than in the general topologies. In this sense, the separation axioms are modified, and in many cases weaker conditions have been adapted for fuzzy topological spaces. With the introduction of fuzzy strongly semi pre-open (short SSPO) sets, we introduce the new separation axioms and investigate their relation with other forms of fuzzy separation axioms. By giving several examples we will be able to show that the newly introduced ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5062 Email addresses: shkumbin.makolli@uni-pr.edu (Sh. Makolli), madob2006@gmail.com (B. Krsteska) https://www.ejpam.com 638 © 2024 EJPAM All rights reserved. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 639 axioms are different from the other fuzzy separation axioms introduced by other authors in [32], [27], and [6]. Compactness is another concept that was also introduced to fuzzy topological spaces. The first ideas of compactness in fuzzy topological spaces were introduced by Chang in [4]. Some other results regarding compact fuzzy spaces were introduced by Goguen [7]. Other authors that have treated compactness in fuzzy topological spaces are Lowen in [15] and [16], Wong [31], T.E. Gantner et al [5], and more recently Saleh S. et al in [26]. The concept of α-compactness was introduced by T.E. Gantner et al in [5] and it is among the most acceptable concepts of compactness in fuzzy topological spaces. With the definition of other forms of generalized fuzzy open sets, different authors, [13], [29], [8], have defined generalization of the concept of fuzzy compactness. In our work, we use the same approach as Gantner et al in [5]. Similarly, we will define the concept of α-SSPO shading and α∗-SSPO shading that are collections of fuzzy sets that constitute only from fuzzy strong semi pre-open sets. Following the introduction of α-SSPO shading (α∗-SSPO shading) we introduce the concepts of α-SSPO compact (α∗-SSPO compact) fuzzy sets and fuzzy spaces. The new concept is stronger then the concept of α-compactness (α∗- compactness). We also investigate the existence of a countable base of fuzzy strong semi pre-open sets, SSPO separability and define the concept of α-SSPO Lindelof sets and fuzzy topological spaces as well as examine their properties. With the definition of fuzzy strong semi pre-continuity and SSPO-irresolute mappings, we investigate the new concept of fuzzy compactness and its properties in relation to the mentioned mappings. Since separation axioms and compactness are among the fundamental principles in the field of Fuzzy Topology, the aim of this research paper is to propose some novel approaches to some theoretical problems in light of new generalized fuzzy opened sets. 2. Preliminaries The concept of fuzzy set was initially formulated by Zadeh in [33]. Chang in [4] introduced the concept of fuzzy topological spaces (short fts). Definition 1. [33] Let X be a space of points (objects). A fuzzy set (class) A in X is characterized by a membership (characteristic) function A(x) which associates with each point in X a real number in the interval [0, 1], with the value of A(x) representing the ”grade of membership” of x in A. In other words, the nearer the value of A(x) to 1, the higher the grade of membership of x in A. Definition 2. [20] Given a fuzzy set A of a fuzzy topological space (X, τ) , the support of the set A is defined as the set suppA = {x ∈ X : A(x) > 0}. Lemma 1. ([1], [22], [12], [13]) Let f : X → Y be a mapping. The following statements hold: (i) ff−1(B) ≤ B, for every fuzzy set B in Y ; (ii) f−1f(A) ≥ A, for every fuzzy set A in X; Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 640 (iii) f(Ac) ≤ (f(A))c, for every fuzzy set A in X ; (iv) f−1(Bc) = (f−1(B))c, for every fuzzy set B in Y ; (v) If A1, A2 are fuzzy sets in X such that A1 ≤ A2, then f(A)1 ≤ f(A)2; (vi) If B1, B2 are fuzzy sets in Y such that B1 ≤ B2, then f−1(B1) ≤ f−1(B2); (vii) If f is an injective mapping, then f−1f(A) = A for every fuzzy set A inX; (viii) If f is a surjective mapping, then ff−1(B) = B for every fuzzy set B in Y ; (ix) If f is a bijective mapping, then f(Ac) = (f(A))c, for every fuzzy set A in X; (x) f( ∧ i∈I Ai) ≤ ∧ i∈I f(Ai), for every family {Ai, i ∈ I} of fuzzy sets from X and I representing a set of indexes; (xi) f( ∨ i∈I Ai) = ∨ i∈I f(Ai), for every family {Ai, i ∈ I} of fuzzy sets from X; (xii) f−1( ∧ i∈I Bi) = ∧ i∈I f −1(Bi), for every family {Bi, i ∈ I} of fuzzy sets from Y and I representing a set of indexes; (xiii) f−1( ∨ i∈I Bi) = ∨ i∈I f −1(Bi), for every family {Bi, i ∈ I} of fuzzy sets from Y ; Definition 3. ([22], [23]) A fuzzy point xα of a fuzzy topological space X is a fuzzy set defined as: xα(z) = { α if z = x 0 if otherwise The support of the fuzzy point xα is only the element x with the value of membership α. If α = 1 then xα is called a singleton. Definition 4. A fuzzy set A of the fuzzy topological space X is called: (i) Fuzzy preopen if and only if A ≤ int(clA) ([3], [27]); (ii) Fuzzy preclosed if and only if Ac is a fuzzy preopen set of a fts X ([3], [14], [27]). Given any fuzzy topological space (X, τ) the family of all fuzzy preopen (preclosed) sets is denoted FPO(τ) (FPC(τ)). Definition 5. Let A be a fuzzy set of a fts (X, τ). Then: (i) pintA = ∧{B ≤ A;B ∈ FPO(τ)}, is called the fuzzy preinterior of the set A ([27]); (ii) pclA = ∨{B ≥ A;B ∈ FPC(τ)}, is called the fuzzy preclosure of the set A ([27]). Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 641 Definition 6. A fuzzy set A of a fts X is called: • Fuzzy strongly preopen (strongly preclosed) if and only if A ≤ int(pclA) (A ≥ cl(pintA)) ([12]); • Fuzzy strongly semi pre-open (strongly semi pre-closed) if and only if A ≤ int(pclA) ∨ pcl(intA) (A ≥ cl(pintA) ∧ pint(clA)) ([17]). The family of all fuzzy strongly preopen (strongly preclosed) sets in (X, τ) is denoted by FSPO(τ) (FSPC(τ)); The family of all fuzzy strongly semi pre-open (strongly semi pre-closed) sets is denoted FSSPO(τ) (FSSPC(τ)). Definition 7. [17] If A is a fuzzy set of a fts X, then: (i) The set: sspintA = ∨{B ≤ A;B ∈ FSSPO(τ)}, is called the fuzzy strong semi preinterior of set A. (ii) The set: sspclA = ∧{B ≥ A;B ∈ FSSPC(τ)}, is called the fuzzy strong semi preclosure of set A. Lemma 2. [17] If A is a fuzzy set of a fuzzy topological space (X, τ), then: • sspclAc = (sspintA)c; • sspintAc = (sspclA)c. Definition 8. Let f : (X, τ) → (Y, δ) be a mapping from a fts (X, τ) to a fts (Y, δ). The mapping f is called: (i) Fuzzy continuous if f−1(B) is a fuzzy open set of X, for each B ∈ δ ([2], [4], [20], [21]); (ii) Fuzzy open (closed) if f(A) is a fuzzy open (closed) set of Y , for each A ∈ τ ([2], [4], [20], [21]); (iii) Fuzzy strong semi pre-continuous if f−1(B) ∈ FSSPO(τ) for every B ∈ δ ([17]); (iv) Fuzzy SSPO-irresolute continuous if f−1(B) ∈ FSSPO(τ) for each B ∈ FSSPO(δ) ([17], [18]); (v) Fuzzy SSPO homeomorphism if it is a bijective mapping and if the mapping f and its inverse are both fuzzy SSPO − irresolute continuous. ([18]). The concepts of fuzzy separation axioms FT0 ( FT1, FTs, FT2, FT2 1 2 , FR, FT3, FN, FT4) will be based on definitions given in [6]. The concepts of α − shading (α∗ − shading), α − subshading (α∗ − subshading) and α− compact (α∗ − compact) will be based on the definitions given in ([5], [13], [19]). Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 642 Definition 9. [5] Let (X, τ) be a fuzzy topological space and let α ∈ [0, 1]. A collection F of fuzzy sets of X is called α − centered (α∗ − centered) if for every finite subcollection U of F , there exist x ∈ X such that U(x) ≥ 1− α (respectively U(x) > 1− α), for every U ∈ U . Definition 10. [30] A fuzzy topological space X is called fuzzy separable if there exists a countable sequence of fuzzy points {xi}i∈N such that for each fuzzy open set A ̸= 0X , there exists xi ∈ A. 3. Axioms of fuzzy strong semi pre-separation Initially, we will define the axioms of fuzzy strong semi pre-separation. Definition 11. A fuzzy topological space X is a fuzzy strong semi pre-T0 (or short FSSPT0) if and only if for every pair of fuzzy points p1 and p2 with different supports, there exists a fuzzy strongly semi pre-open set O such that p1 ≤ O ≤ pc2 or p2 ≤ O ≤ pc1. It follows directly from the Definition 11 and [6] that every FT0 space is also an FSSPT0 while the converse is not true in general. We will give the following example to illustrate this fact. Example 1. Given a set X = {p1, p2}, fuzzy set U = {(p1, 0), (p2, 0.6)} and fts τ = {0, U, 1}. It is obvious that the fts is not FT0 but it is an FSSPT0 since for O = {(p1, 0), (p2, 1)}, O ∈ FSSPO(τ) it follows that p2 ≤ O ≤ pc1. Theorem 1. If the fuzzy topological space X is an FSSPT0, and given any pair of fuzzy singletons p1 and p2 with different supports, then sspclp1 ̸= sspclp2. Proof. Since the fuzzy topological space (X, τ) is an FSSPT0, then given two fuzzy singletons p1 and p2 with different support, it is obvious that there exist a set O ∈ FSSPO(τ), such that p1 ≤ O ≤ pc2. If we use the fact that sspclp2 ≤ Oc and since p1 ≰ Oc, it follows that sspclp1 ̸= sspclp2. If we refer to Example 1, it is obvious that sspclp1 ≤ {(p1, 1), (p2, 0.4)} while sspclp2 = 1X , that is sspclp1 ̸= sspclp2. Definition 12. A fuzzy topological space X is a fuzzy strong semi pre-T1 (or short FSSPT1) if and only if for any pair of fuzzy points p1 and p2 which have different sup- ports, there exist fuzzy strongly semi pre-open sets O1, O2 such that p1 ≤ O1 ≤ pc2 and p2 ≤ O2 ≤ pc1. We can formulate and prove the following theorem which gives some characteristic properties for FSSPT1 spaces. Theorem 2. The fuzzy topological space X is an FSSPT1 space if and only if each fuzzy singleton is a fuzzy strongly semi pre-closed set. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 643 Proof. Let us suppose that the given fuzzy topological space X is an FSSPT1 space. If we consider fuzzy singletons p and x with different support, it is obvious that there exist fuzzy strongly semi pre-open sets Op and Ox such that p ≤ Op ≤ xc and x ≤ Ox ≤ pc. Now, if we consider the fuzzy set pc as a fuzzy set that contains all of its fuzzy points, we can write that as pc = ∨ x≤pc x ≤ ∨ x≤pc Ox. Since X is an FSSPT1 space, we also have that: Ox ≤ pc =⇒ ∨ x≤pc Ox ≤ pc From the two last inequalities, we have pc = ∨ x≤pc Ox. In other words, the fuzzy set pc is a fuzzy strongly semi pre-open set as a union of such sets and subsequently the singleton p is a fuzzy strongly semi pre-closed set. Conversely, if each fuzzy singleton of a fuzzy topological space X is a fuzzy strongly semi pre-closed set and if we consider any pair of fuzzy singletons p and x with different support, it is obvious that pc and xc are fuzzy strongly semi pre-open sets such that p ≤ xc and x ≤ pc. If we write xc = Op and pc = Ox we get the following p ≤ Op ≤ xc and x ≤ Ox ≤ pc, which means that fuzzy topological space X is an FSSPT1. Corollary 1. A fuzzy topological space is an FSSPT1 space if and only if for each pair of fuzzy singletons p1 and p2 which have different supports, there exist fuzzy strongly semi pre-open sets O1, O2 such that O1(p1) = 1, O1(p2) = 0 and O2(p1) = 0, O2(p2) = 1. Proof. If the fuzzy topological space is an FSSPT1 space, then according to Theorem 2, the conditions are met if we put pc2 = O1 and pc1 = O2. Conversely, if for any pair of fuzzy singletons p1 and p2, with different supports, there exist fuzzy strong semi pre-open sets O1, O2 such that O1(p1) = 1, O1(p2) = 0 and O2(p1) = 0, O2(p2) = 1, it is obvious that p1 ≤ O1 ≤ pc2 and p2 ≤ O2 ≤ pc1, which means that the fuzzy topological space is an FSSPT1 space. We can easily conclude that any FSSPT1 space is also an FSSPT0 while the converse is not always true. If we consider example 1, it is obvious that the fuzzy topological space (X, τ) is not an FSSPT1 space. Definition 13. A fuzzy topological space X is a fuzzy strong semi pre-Ts (or short FSSPTs) if and only if every fuzzy point is a fuzzy strongly semi pre-closed set. By Theorem 2 it is obvious that any FSSPTs space is also an FSSPT1. With the following example, we will show that the converse is not always true. Example 2. Given a set X = {p, q}, and fuzzy sets U = {(p, 1), (q, 0)} , V = {(p, 0), (q, 1)}, the fuzzy topological space τ = {0, U, V, 1} is FSSPT1 but it is not FSSPTs. It is obvious that every singleton is a fuzzy strongly semi pre-closed set, and the conclusion follows from Theorem 2. Example 3. Given a set X = {p1, p2}, and fuzzy sets U = {(p1, 0.6), (p2, 0)}, V = {(p1, 0.7), (p2, 0)} and W = {(p1, 0.8), (p2, 0.7)}. If τ = {0, U, V,W, 1}. It can be shown that the fuzzy topological space (X, τ) is FSSPT0 but it is not FSSPT1 and FSSPTs. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 644 Definition 14. A fuzzy topological space X is a fuzzy strong semi pre-Hausdorff (or short FSSPT2) if and only if for any pair of fuzzy points p1 and p2, which have different supports, there exist fuzzy strongly semi preo-pen sets O1, O2 such that p1 ≤ O1 ≤ pc2, p2 ≤ O2 ≤ pc1 and O1 ≤ Oc 2. Theorem 3. The fuzzy topological space (X, τ) is an FSSPT2 if and only if there exists a fuzzy strongly semi pre-open set O such that p1 ≤ O ≤ sspclO ≤ pc2, where p1 and p2 are any pair of fuzzy points from X that have different supports. Proof. If the fuzzy topological space (X, τ) is an FSSPT2 then it is obvious that for any pair of fuzzy points p1, p2 there must exist a set O ∈ FSSPO(τ), p1 ≤ O ≤ pc2, and a set W ∈ FSSPO(τ) such that p2 ≤ W ≤ pc1 and O ≤ W c. It follows that p1 ≤ O ≤ sspclO ≤ sspclW c = W c ≤ pc2. Conversely, if we denote by (sspclO)c = W , it is obvious that p2 ≤ W and W ∈ FSSPO(τ). Now we have a case where p1 ≤ O ≤ pc2, p2 ≤ W ≤ pc1 and also O ≤ W c, meaning that (X, τ) is an FSSPT2. Example 4. Given a set X = {p1, p2}, and fuzzy sets U = {(p1, 0.6), (p2, 0)}, V = {(p1, 0), (p2, 0.6)} and W = {(p1, 0.6), (p2, 0.8)}. If τ = {0, U, V, U ∨ V,W, 1}, it can be shown that the fuzzy topological space (X, τ) is an FSSPT2 and it is not and FSSPTs. It is also obvious that this is an example of a fuzzy space that is not an FT2 and not an FSPT2 ([13]). Example 5. Let X be an infinite set and let the family of fuzzy sets be defined as: τ = {G|suppGc is a finite set}. It is clear that (X, τ) is a fuzzy topological space and that each fuzzy point in τ is a fuzzy strongly semi pre-closed set. From the other perspective, it is impossible to find any pair of fuzzy points p1, p2 and fuzzy strongly semi pre-open sets O1, O2 such that p1 ≤ O1 ≤ pc2, p2 ≤ O2 ≤ pc1 and O1 ≤ Oc 2, because the last relation would imply that an infinite fuzzy set is the subset of a finite fuzzy set. The first argument shows that the fuzzy topological space (X, τ) is an FSSPTs while the second arguments tells us that it is not an FSSPT2 space. In other words, we have illustrated with two last examples that the classes of FSSPT2 spaces and FSSPTs spaces are independent. Definition 15. A fuzzy topological space X is a fuzzy strong semi pre-Urysohn (or short FSSPT2 1 2 ) if and only if for any pair of fuzzy points p1 and p2, which have different supports, there exist fuzzy strongly semi pre-open set O1, O2 such that p1 ≤ O1 ≤ pc2, p2 ≤ O2 ≤ pc1 and sspclO1 ≤ (sspclO2) c. According to the above definition, it is clear that any fuzzy strong semi pre-Urysohn space is also a fuzzy strong semi pre-Hausdorff space. With the following example, we will illustrate that the converse statement does not hold in general case. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 645 Example 6. Let X be an infinite set and let there p0 be a fuzzy point with x0 ∈ X as its support. Let us define the family of fuzzy sets as follows: A = {O|O(x0) ≤ p0(x0)} B = {O|suppOc is a finite set} It is obvious that τ = A∨B is a fuzzy topological space that is a case of an FSSPT2 space which is not an FSSPT2 1 2 . Definition 16. A fuzzy topological space X is a fuzzy strong semi pre-regular (or short FSSPR) if and only if for every fuzzy points p and every fuzzy strongly semi pre-closed set C in X such that p ≤ Cc, there exist fuzzy strongly semi pre-open sets O1, O2 such that p ≤ O1, C ≤ O2 and O1 ≤ Oc 2. The definition of FSSPR spaces can also be given in the equivalent form as it follows. The Fuzzy topological space (X, τ) is an FSSPR space if and only if for every fuzzy point p and every fuzzy strongly semi pre-open set O such that p ≤ O, there exists a fuzzy strongly semi pre-open set U such that p ≤ U ≤ sspclU ≤ O. Definition 17. A fuzzy topological space X which is an FSSPR and FSSPTs is called FSSPT3. We can give also a different and weaker condition of fuzzy strong semi pre-regularity with the following definition. Definition 18. A fuzzy topological space X is a fuzzy strong semi pre-weakly regular (or short FSSPWR) if and only if for every fuzzy points p and every fuzzy closed set C in X such that p ≤ Cc, there exist fuzzy strongly semi pre-open sets O1, O2 such that p ≤ O1, C ≤ O2 and O1 ≤ Oc 2. Theorem 4. Let X be an FSSPR space, then for every fuzzy strongly semi pre-closed set F in X and any fuzzy point p ≤ F c, there exist fuzzy strongly semi pre-open sets U,W such that p ≤ U , F ≤ W and sspclU ≤ (sspclW )c. Proof. According to the statement of the theorem, for every fuzzy point p ≤ F c and the fact that X is an FSSPR space, there exist fuzzy strongly semi pre-open sets O,W such that p ≤ O, F ≤ W and O ≤ W c. Also from the equivalent definition of FSSPR spaces, for every fuzzy point p and a fuzzy strongly semi pre-open set O such that p ≤ O, there exists a fuzzy strongly semi pre-open set U such that p ≤ U ≤ sspclU ≤ O. Now the conclusion of the theorem is obvious. Corollary 2. Every FSSPT3 space is an FSSPT2 1 2 space. Proof. It follows directly from the previous theorem and from the fact that an FSSPT3 space is an FSSPR and FSSPTs space. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 646 Theorem 5. Any fuzzy topological space (X, τ) which is an FSSPR and FSSPT0 space is also an FSSPT2 1 2 space. Proof. We are going to take into consideration two fuzzy points p1 and p2 with different support. Based on the assumption that (X, τ) is an FSSPT0 space, it follows that there exists a set U ∈ FSSPO(τ) such that p1 ≤ U ≤ pc2. If we denote by F = U c, F ∈ FSSPC(τ), it is obvious that p1 ≤ F c and since (X, τ) is an FSSPR space, it implies the existence of fuzzy sets V,W ∈ FSSPO(τ) such that p1 ≤ V , F ≤ W , and V ≤ W c. Now, from the Theorem 4 we also have that sspclV ≤ (sspclW )c and combining it with the fact that p2 ≤ U c = F ≤ W and as well as p1 ≤ V , we reach the desired result. The latest conditions imply that for any given pair of fuzzy points p1 and p2 with different support, there exist fuzzy sets V,W ∈ FSSPO(τ) such that p1 ≤ V ≤ pc2 , p2 ≤ W ≤ pc1 and sspclV ≤ (sspclW )c, hence the fuzzy topological space (X, τ) is FSSPT2 1 2 . Definition 19. A fuzzy topological space X is a fuzzy strong semi pre-normal (or short FSSPN) if and only if for every pair of fuzzy strongly semi pre-closed sets C1, C2, such that C1 ≤ Cc 2, there exist fuzzy strongly semi pre-open sets O1, O2 such that C1 ≤ O1, C2 ≤ O2 and O1 ≤ Oc 2. A fuzzy topological space with FSSPN and FSSPTs properties is called an FSSPT4 space. Clearly, any FSSPT4 space is also an FSSPT3 space. We can formulate the following theorem which gives a necessary and sufficient condition for the existence of FSSPN spaces. Theorem 6. The fuzzy topological space (X, τ) is an FSSPN if and only if for any F ∈ FSSPC(τ) and a fuzzy set O ∈ FSSPO(τ) such that F ≤ O, there exists a fuzzy set W ∈ FSSPO(τ) such that F ≤ W ≤ sspclW ≤ O. Proof. We can use similar argumentation as in Theorem 4. Definition 20. A fuzzy topological space X is a fuzzy strong semi pre-weakly normal (or short FSSPWN) if and only if for every pair of fuzzy strongly semi pre-closed sets C1, C2, such that C1 ∧ C2 = ∅, there exist fuzzy strongly semi pre-open sets O1, O2 such that C1 ≤ O1, C2 ≤ O2 and O1 ≤ Oc 2. We can formulate the following theorem in regards to the FSSPWN spaces. Theorem 7. Every fuzzy topological space (X, τ) which is an FSSPN space is also an FSSPWN space. Proof. In fuzzy topological spaces the following implication is always true: C1 ∧ C2 = ∅ =⇒ C1 ≤ Cc 2 In general, the equivalence is not always valid for fuzzy sets, this means that if the fuzzy topological space is FSSPN then it is also an FSSPWN . Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 647 4. Axioms of fuzzy strong semi pre-separation and fuzzy strong semi pre-continuous mappings In this section, we will investigate the relation between fuzzy separation axioms, fuzzy pre-separation axioms and different forms of fuzzy continuity. Theorem 8. Let f : X → Y be a fuzzy strong semi pre-continuous and injective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space Y is an FT2 (FT1, FT0) space then X is an FSSPT2 (FSSPT1, FSSPT0) space. Proof. Let us suppose that fuzzy points p, q ≤ X represent any pair of fuzzy points with different support. According to the assumption of the theorem, the mapping f : X → Y is an injective mapping, it is obvious that f(p), f(q) are two fuzzy points in Y with different support. Now, since the fuzzy topological space Y is an FT2, there exist fuzzy open sets U, V such that: f(p) ≤ U ≤ f(q)c, f(q) ≤ V ≤ f(p)c and U ≤ V c. Since the mapping f is a fuzzy strong semi pre-continuous mapping then f−1(U), f−1(V ) are two fuzzy strongly semi pre-open sets in X such that: p ≤ f−1(U) ≤ qc, q ≤ f−1(V ) ≤ pc and also f−1(U) ≤ f−1(V )c, that is, the fuzzy topological space X is an FSSPT2 space. Similarly, we can prove the cases when Y is an FT1 and FT0 space. Theorem 9. Let f : X → Y be a fuzzy strong semi pre-open and bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space X is an FT2 (FT1, FT0) space then Y is an FSSPT2 (FSSPT1, FSSPT0) space. Proof. Let us suppose that p, q ≤ Y are two fuzzy points with different support. It is obvious from the conditions of the theorem that f−1(p), f−1(q) ≤ X are two fuzzy points with different support. Since the fuzzy topological space X is an FT2, there are fuzzy open sets U, V such that: f−1(p) ≤ U ≤ f−1(q)c, f−1(q) ≤ V ≤ f−1(p)c and U ≤ V c. Based on the assumption of the theorem, the images f(U), f(V ) of U and V are fuzzy strongly semi pre-open sets in Y and the following stands: p ≤ f(U) ≤ qc, q ≤ f(V ) ≤ pc and f(U) ≤ f(V )c, which means that Y is an FSSPT2 space. In similar manner we can show that the same holds when X is an FT1 and FT0 space. Theorem 10. Let f : X → Y be a fuzzy strong semi pre-continuous and injective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space Y is an FTs space then X is an FSSPTs space. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 648 Proof. Let p be any fuzzy point in the fuzzy topological space X then f(p) is a fuzzy point in Y . Since Y is an FTs space, it means that any fuzzy point is a fuzzy closed set, that is f(p) is a fuzzy closed set in Y . Because f is a fuzzy strong semi pre-continuous mapping and it is an injective mapping then f−1(f(p)) = p, and p is a fuzzy strongly semi pre-closed set in X. Since p is any fuzzy point of X, that means that the fuzzy topological space X is an FSSPTs. Theorem 11. Let f : X → Y be a fuzzy strongly semi pre-open and bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space X is an FTs space then Y is an FSSPTs space. Proof. Similar to Theorem 10. Theorem 12. Let f : X → Y be a fuzzy strong semi pre-continuous and injective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space Y is an FT2 1 2 space then X is an FSSPT2 1 2 space. Proof. Let us suppose that p, q ≤ X are two fuzzy points with different support. Since f : X → Y is a fuzzy strong semi pre-continuous and an injective mapping, it follows that f(p), f(q) ≤ Y are two fuzzy points with different support. Due to the fact that the fuzzy topological space Y is an FT2 1 2 , there are fuzzy open sets U, V such that f(p) ≤ U ≤ f(q)c, f(q) ≤ V ≤ f(p)c and clU ≤ (clV )c. Based on the assumption of the theorem, the images f−1(U), f−1(V ), of U and V are fuzzy strongly semi pre-open sets in Y and p ≤ f−1(U) ≤ qc, q ≤ f−1(V ) ≤ pc. According to the Theorem 4.1. [17][17] we have that: sspclf−1(U) ≤ f−1(clU) ≤ f−1(clV )c ≤ f−1(intV c) ≤ sspintf−1(V c) ≤ (sspclf−1(V ))c The last expression can also be summarized as sspclf−1(U) ≤ (sspclf−1(V ))c which means that X is an FSSPT2 1 2 space. Theorem 13. Let f : X → Y be a fuzzy strongly semi pre-open and bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space X is an FT2 1 2 space then Y is an FSSPT2 1 2 space. Proof. Let us suppose that p, q ≤ Y are two fuzzy points with different support. It is obvious from the conditions of the theorem that f−1(p), f−1(q) ≤ X are two fuzzy points with different support. Since the fuzzy topological space X is an FT2 1 2 , there are fuzzy open sets U, V such that f−1(p) ≤ U ≤ f−1(q)c, f−1(q) ≤ V ≤ f−1(p)c and also clU ≤ (clV )c. Based on the assumption of the theorem, the images f(U), f(V ) of U and V are fuzzy strongly semi pre-open sets in Y and the following stands: p ≤ f(U) ≤ qc, q ≤ f(V ) ≤ pc and Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 649 sspclf(U) ≤ f(clU) ≤ f(clV )c ≤ (sspclf(V ))c which means that Y is an FSSPT2 1 2 space. Theorem 14. Let f : X → Y be a fuzzy closed and fuzzy strong semi pre-continuous and bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space Y is an FR space then X is an FSSPWR space. Proof. Proof: Let X be a fuzzy topological space and let p be a fuzzy point, let F be any fuzzy closed set in X such that p ≤ F c. Then, f(p) is a fuzzy point in Y and according to the conditions of the theorem f(p) ≤ f(F )c. It is obvious that the fuzzy set f(F ) is a fuzzy closed set in Y . Since Y is an FR space, then there exist fuzzy open sets U, V such that f(p) ≤ U , f(F ) ≤ V and U ≤ V c. If we refer again to the conditions of the theorem, then we have: p ≤ f−1(U), F ≤ f−1(V ) and f−1(U) ≤ f−1(V )c. It is obvious that f−1(U) and f−1(V ) are fuzzy strongly semi pre-open sets in X. Hence the fuzzy topological space X is an FSSPWR space. Theorem 15. Theorem Let f : X → Y be a fuzzy continuous and fuzzy strongly semi pre-open and bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space X is an FR space then Y is an FSSPWR space. Proof. Similar to Theorem 14. Theorem 16. Let f : X → Y be a fuzzy SSPO-irresolute and injective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space Y is an FSSPT2 1 2 (FSSPT2, FSSPTs, FSSPT1, FSSPT0) space then X is also an FSSPT2 1 2 (FSSPT2, FSSPTs, FSSPT1, FSSPT0) space. Proof. Let us suppose that fuzzy points p, q ≤ X represent any pair of fuzzy points with different support. According to the assumption of the theorem, the mapping f : X → Y is an injective mapping, it is obvious that f(p), f(q) are two fuzzy points in Y with different support. Now, since the fuzzy topological space Y is an FSSPT2 1 2 , there exist fuzzy strongly semi pre-open sets U, V such that: f(p) ≤ U ≤ f(q)c, f(q) ≤ V ≤ f(p)c and sspclU ≤ (sspclV )c. Since the mapping f is a fuzzy SSPO-irresolute, it follows that f−1(U), f−1(V ) are two fuzzy strongly semi pre-open sets in X such that: p ≤ f−1(U) ≤ qc, q ≤ f−1(V ) ≤ pc. From the conditions set out by Theorem 1 in [18], we can prove that: Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 650 sspclf−1(U) ≤ f−1(sspclU) ≤ f−1((sspclV )c) = f−1(sspintV c) ≤ ≤ sspint(f−1(V c)) = (sspcl(f−1(V ))c. We have shown that for the given fuzzy strongly semi pre-open sets f−1(U), f−1(V ) it follows that sspclf−1(U) ≤ (sspcl(f−1(V ))c, which means that the fuzzy topological space X is an FSSPT2 1 2 space. In the similar way we can prove the cases when Y is an FSSPT2, FSSPTs, FSSPT1, FSSPT0 space. Theorem 17. Let f : X → Y be a fuzzy SSPO-irresolute open and bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space X is an FSSPT2 1 2 (FSSPT2, FSSPTs, FSSPT1, FSSPT0) space then Y is also an FSSPT2 1 2 (FSSPT2, FSSPTs, FSSPT1, FSSPT0) space. Proof. Let us show only the case when the fuzzy topological space X is an FSSPTs space. Other cases are proved in similar manner (similar to Theorem 16). Let q ≤ Y be any fuzzy point of Y . The preimage of this point satisfies the following f−1(q) ≤ X. Based on the conditions of the theorem and from the fact that X is an FSSPTs space, that is, any fuzzy point f−1(q) ≤ X is a fuzzy strongly semi pre-closed set in X. It follows that the image f(f−1(q)) = q ≤ Y of the fuzzy point f−1(q) is also a fuzzy strongly semi pre-closed set in Y . In other words any fuzzy point of Y is also a fuzzy strongly semi pre-closed set and therefore Y is also an FSSPTs space. Theorem 18. Let f : X → Y be a fuzzy SSPO-irresolute closed and fuzzy strong semi pre-continuous bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space Y is an FSSPN (FSSPWN,FSSPT3, FSSPR) space then X is also an FSSPN (FSSPWN,FSSPT3, FSSPR) space. Proof. Let F1, F2 be two fuzzy strongly semi pre-closed sets in X such that F1 ≤ F c 2 . Obviously, due to the conditions of the theorem, f(F1), f(F2) are two fuzzy strongly semi pre-closed sets in Y such that f(F1) ≤ (f(F2)) c. Since Y is an FSSPN space, there are fuzzy strongly semi pre-open sets W1,W2 such that: f(F1) ≤ W1, f(F2) ≤ W2 and W1 ≤ W c 2 . From the assumption that f is a fuzzy strong semi pre-continuous mapping, it follows that f−1(W1) and f−1(W2) are two fuzzy strongly semi pre-open sets in X and the following stands: F1 ≤ f−1(W1), F2 ≤ f−1(W2) and f−1(W1) ≤ (f−1(W2)) c Therefore the fuzzy topological space X is an FSSPN space. Similarly, we can prove the other cases when Y is an FSSPWN,FSSPT3 and FSSPR space. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 651 Theorem 19. Let f : X → Y be a fuzzy SSPO-irresolute open and fuzzy strong semi pre-continuous bijective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy topological space X is an FSSPN (FSSPWN,FSSPT3, FSSPR) space then Y is also an FSSPN (FSSPWN,FSSPT3, FSSPR) space. Proof. Let V1, V2 be two fuzzy strongly semi pre-closed sets in Y such that V1 ≤ V c 2 . From the assumptions of the theorem, since f is a fuzzy strong semi pre-continuous mapping, it follows that f−1(V1), f −1(V2) are two fuzzy strongly semi pre-closed sets in X such that f−1(V1) ≤ f−1(V1) c. Now, since X is an FSSPN space, there are fuzzy strongly semi pre-open sets U1, U2 such that: f−1(V1) ≤ U1, f −1(V2) ≤ U2 and U1 ≤ U c 2 . From the assumption that f is a fuzzy SSPO-irresolute open mapping, it follows that f(U1) and f(U2) are fuzzy strongly semi pre-open sets in Y and the following conditions are fulfilled: V1 ≤ f(U1), V2 ≤ f(U2) and f(U1) ≤ f(U2) c. Hence the fuzzy topological space Y is an FSSPN space. In similar way we can prove the other cases when the fuzzy topological space X is an FSSPWN,FSSPT3, and FSSPR space. 5. A novel form of fuzzy compactness In this section, we will introduce a novel form of compactness in fuzzy topological spaces. The properties of this new form of fuzzy compactness, similarities, and differences with other forms of fuzzy compactness will also be investigated. We will initially give the following definition. Definition 21. Let (X, τ) be a fuzzy topological space and let α ∈ [0, 1]. A collection S of fuzzy strongly semi pre-open sets of (X, τ) is called an α− SSPO shading (respectively α∗−SSPO shading) of the fuzzy set A if, for every a ∈ suppA, there exist a set W ∈ S such that W (a) > α (respectively W (a) ≥ α). A subcollection C of sets from α−SSPO shading (respectively α∗−SSPO shading) S which is also an α−SSPO shading (respectively α∗− SSPO shading) for the given fuzzy set A, is called an α−SSPO subshading (respectively α∗ − SSPO subshading) of the collection S. With the concept of α−SSPO shading ( α∗−SSPO shading), which are analogous to the concept of open covers in the ordinary topology, we can define the concept of α−SSPO compactness. Definition 22. The fuzzy set A of the fuzzy topological space (X, τ) is called α− SSPO compact (α∗−SSPO compact) if every α−SSPO shading (respectively α∗−SSPO shad- ing) of the set A has a finite α−SSPO subshading (respectively α∗−SSPO subshading). If instead of any set A we consider the set X in general, then we can state that space (X, τ) is α− SSPO compact (α∗ − SSPOcompact). Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 652 Definition 23. The fuzzy set A of the fuzzy topological space (X, τ) is called countable α − SSPO compact (respectively countable α∗ − SSPO compact) if every countable α − SSPO shading (respectively α∗ − SSPO shading) of the set A has a finite α − SSPO subshading (respectively α∗ − SSPO subshading). If instead of the fuzzy set A we consider the space X then we can state that the fuzzy topological space X is a countable α− SSPO compact (respectively countable α∗ − SSPO compact). From definition 23 it is obvious that if the fuzzy topological space X is α − SSPO compact (α∗ − SSPO compact) then it is also countable α− SSPO compact (countable α∗ − SSPO compact). Directly from the definition, we can conclude that any fuzzy point is α− SSPO compact set and α∗ − SSPO compact set. It is also obvious that any fuzzy set in X is 1− SSPO compact and 0∗−SSPO compact. Also from the definition 22 it follows that any α−SSPO compact (α∗ − SSPO compact) space is also an α- compact (α∗-compact) space. The converse is not always true as it can be presented with the following example. Example 7. If X is any infinite set and if α ∈ [0, 1], for any p ∈ X we will define the following sets: Uα p (x) = { 1 if x = p α if x ̸= p Let as denote with Tα the fuzzy topology on X which is generated by {Uα p (x) : p ∈ X}. In [5] it was shown that (X, Tα) is β-compact for β = 1 or 0 ≤ β < α and is β∗-compact for 0 ≤ β ≤ α. It is obvious that (X, Tα) is β − SSPO compact only for β = 1 or 0 ≤ β < α and is β∗−SSPO compact for 0 ≤ β ≤ α. Moreover, if γ < α, then (X, Tγ) is α-compact and α∗-compact, see [5], but it is neither α− SSPO compact nor α∗ − SSPO compact. Theorem 20. The fuzzy topological space (X, τ) is α−SSPO compact (respectively α∗− SSPO compact) if and only if for every α-centered (α∗-centered) family F consisting of fuzzy strongly semi pre-closed sets in (X, τ), there exists x ∈ X such that F (x) ≥ 1 − α (F (x) > 1− α), for every F ∈ F . Proof. Let us suppose that F is an α-centered family consisting of fuzzy strongly semi pre-closed sets in (X, τ) such that for each x ∈ X, there exists a set F ∈ F such that F (x) < 1−α. Then the family of sets W = {F c, F ∈ F} is an α−SSPO shading of (X, τ) and it is evident that it does not have a finite α − SSPO subshading. If it had a finite α − SSPO subshading F c 1 , F c 2 , . . . , F c k , then due to the fact that F is α-centered there exists x ∈ X, such that Fj(x) ≥ 1 − α for all j = 1, 2, . . . , k and consequently F c j (x) ≤ α for all j = 1, 2, . . . , k. Conversely, let us suppose that family S of fuzzy strongly semi pre-open sets of (X, τ) is an α-SSPO shading of X and that it has no finite α-SSPO subshading. Then the collection of fuzzy strongly semi pre-closed sets F = {Sc, S ∈ S} is α-centered because for Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 653 Sc 1, S c 2, . . . , S c k ∈ F there must exist x ∈ X such that Sj(x) ≤ α for all j = 1, 2, . . . , k (or otherwise the family S has a finite α − SSPO subshading), and therefore Sc j (x) ≥ 1 − α for all j = 1, 2, . . . , k. On the other side, given any x ∈ X there exists S ∈ S such that S(x) > α and consequently Sc ∈ F and as well Sc(x) < 1− α. In the same manner we can prove the case when the fuzzy topological space (X, τ) is α∗ − SSPO compact. Corollary 3. The fuzzy topological space (X, τ) is α− SSPO compact (respectively α∗ − SSPO compact) if and only if for every α-centered (α∗-centered) family F consisting of fuzzy sets in (X, τ), there exists x ∈ X such that sspclF (x) ≥ 1− α (sspclF (x) > 1− α), for every F ∈ F . Proof. Follows directly from Theorem 20. Theorem 21. The fuzzy topological space (X, τ) is countable α−SSPO compact (respec- tively countable α∗−SSPO compact) if and only if for every countable α-centered (count- able α∗-centered) family F consisting of fuzzy strongly semi pre-closed sets in (X, τ), there exists x ∈ X such that F (x) ≥ 1− α (F (x) > 1− α), for every F ∈ F . Proof. Similar to Theorem 20 Theorem 22. Let A be an α−SSPO compact (α∗ −SSPO compact) fuzzy set in (X, τ) and let B ∈ FSSPC(τ), then the fuzzy set A ∧B is an α− SSPO compact (α∗ − SSPO compact) fuzzy set in the fuzzy topological space (X, τ). Proof. Let us suppose that U = {Ui, i ∈ I} is an α − SSPO shading of the fuzzy set A∧B. It follows that the collection of sets {Ui, i ∈ I}∨Bc is an α−SSPO shading of the fuzzy set A. The last is true because if a ∈ suppA then a ∈ supp(A ∧ B) or B(a) = 0. If a ∈ supp(A∧B) then there exists Uj ∈ U such that Uj(a) > α, otherwise, if B(a) = 0 then Bc(a) = 1 > α. In other words the collection {Ui, i ∈ I} ∨ Bc is an α − SSPO-shading of the fuzzy set A and since A is α − SSPO compact, there exists a finite α − SSPO subshading {Ui, i = 1, 2, . . . , k} ∨ Bc. It is evident that {Ui, i = 1, 2, . . . , k} is a finite α− SSPO subshading of A ∧B, that is A ∧B is α− SSPO compact. In similar way we can show the case when the fuzzy set A is α∗ − SSPO compact. Corollary 4. Let X be an α − SSPO-compact (α∗ − SSPO compact) fuzzy topological space, then any fuzzy set B ∈ FSSPC(τ) is an α−SSPO-compact (α∗−SSPO-compact) fuzzy set in the fuzzy topological space (X, τ). Proof. It is obvious, from Theorem 22, if we substitute the fuzzy α− SSPO-compact set A with X. Theorem 23. Let A,B be α−SSPO-compact (α∗−SSPO compact) fuzzy sets in (X, τ), then the fuzzy set A ∨ B is also an α − SSPO-compact (α∗ − SSPO compact) fuzzy set in the fuzzy topological space (X, τ). Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 654 Proof. Let us suppose that {Wi, i ∈ I} is an α − SSPO shading of the fuzzy set A∨B. It follows that {Wi, i ∈ I} is also an α−SSPO shading of the fuzzy sets A and B. According to the assumption of the theorem, A and B are two α− SSPO compact fuzzy sets in (X, τ), therefore there exists a finite α−SSPO subshading Wi1 ,Wi2 , ...,Wik of A as well as a finite α−SSPO subshading Wj1 ,Wj2 , ...,Wjm of B. Now if we consider the finite collection of sets Wi1 ,Wi2 , ...,Wik ,Wj1 ,Wj2 , ...,Wjm it is obvious that it consists a finite α−SSPO subshading of {Wi, i ∈ I} and obviously A∨B is an α−SSPO compact fuzzy set in (X, τ). The latter is true since for any x ∈ supp(A ∨ B) = x ∈ (suppA ∪ suppB), there exists Wx ∈ {Wi1 ,Wi2 , ...,Wik ,Wj1 ,Wj2 , ...,Wjm} such that Wx(x) > α. In similar way we can show the case when the fuzzy sets A,B are α∗−SSPO compact. Corollary 5. Let A be a fuzzy set in fuzzy topological space (X, τ). If the fuzzy set A has a finite support then A is an α − SSPO compact (α∗ − SSPO compact) fuzzy set in (X, τ). Corollary 6. Let X be a finite fuzzy topological space, then X is an α− SSPO compact (α∗ − SSPO compact) fuzzy set in (X, τ). Theorem 24. If f : X → Y is a fuzzy SSPO-irresolute mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy set A is α−SSPO compact (α∗−SSPO compact) fuzzy set in X then f(A) is an α−SSPO compact (α∗ −SSPO compact) fuzzy set in Y . Proof. Let us suppose that U = {Ui, i ∈ I} is an α − SSPO shading of the fuzzy set f(A) in Y . Then the family W = {f−1(Ui), Ui ∈ U} is a collection of fuzzy strongly semi pre-open sets of X. Since for every x ∈ suppA we have that f(x) ∈ f(suppA) = suppf(A) and since {Ui, i ∈ I} is an α − SSPO shading of f(A), there exists Uj ∈ U such that Uj(f(x)) > α and subsequently f−1(Uj)(x) = Uj(f(x)) > α, which means that the family W is an α − SSPO shading of A. Since A is α − SSPO compact it follows that W contains a finite α − SSPO subshading, {f−1(Ui), i ∈ J}, where J is a finite set of indexes. Therefore we can conclude that the finite collection {Ui, i ∈ J} is an α− SSPO subshading of the α − SSPO shading U . This is true due to the fact that for every y ∈ suppf(A) there exists x ∈ suppA such that f(x) = y. Now, since {f−1(Ui), i ∈ J} is a finite α − SSPO subshading of A, there exists m ∈ J such that f−1(Um)(x) > α and therefore f−1(Um)(x) = Um(f(x)) = Um(y) > α. We showed that f(A) is an α − SSPO compact set in Y . The other case is proven in a similar way. Corollary 7. If f : X → Y is a fuzzy SSPO-irresolute mapping from the fuzzy topological space X to a fuzzy topological space Y . If X is α−SSPO compact (α∗−SSPO compact) then f(X) is an α− SSPO compact (α∗ − SSPO compact) fuzzy set in Y . Proof. It follows directly from Theorem 24. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 655 Corollary 8. Let f : X → Y is a fuzzy SSPO-irresolute and surjective mapping from the fuzzy topological space X to a fuzzy topological space Y . If X is α − SSPO compact (α∗ − SSPO compact) then Y is an α− SSPO compact (α∗ − SSPO compact) fuzzy set. Proof. It follows from Corollary 7 and since f(X) = Y when f is a surjective mapping. Theorem 25. Let f : X → Y be a fuzzy strong semi pre-continuous and surjective mapping from the fuzzy topological space X to a fuzzy topological space Y . If X is α − SSPO compact (α∗ − SSPO compact) then Y is an α-compact (α∗-compact). Proof. Let us suppose that U = {Ui, i ∈ I} is an α-shading of Y . Then the family W = {f−1(Ui), Ui ∈ U} is a collection of fuzzy strongly semi pre-open sets of X. Since for every x ∈ X we have that f(x) ∈ f(suppX) = suppY and since {Ui, i ∈ I} is an α- shading of Y there exists a fuzzy open set Uj ∈ U such that Uj(f(x)) > α and subsequently f−1(Uj)(x) = Uj(f(x)) > α, which means that the familyW is an α−SSPO shading ofX. Since X is α−SSPO compact it follows that W contains a finite α−SSPO subshading, {f−1(Ui), i ∈ K}, where K is a finite set of indexes. Therefore we can conclude that the finite collection {f(f−1(Ui) = Ui, i ∈ K} is an α-subshading of the α-shading U . The last stands because f is a surjective mapping and due to the fact that for every y ∈ Y there exists x ∈ X such that f(x) = y. Now, since {f−1(Ui), i ∈ K} is a finite α−SSPO subshading of X, there exists m ∈ K such that f−1(Um)(x) > α and therefore f−1(Um)(x) = Um(f(x)) = Um(y) > α. We showed that for any α-shading U = {Ui, i ∈ I} of Y there exists a finite α-subshading {Ui, i ∈ K} and hence Y is an α-compact set. The proof of the case when X is α∗ − SSPO compact is similar and is therefore omitted. Theorem 26. Let the mapping f : X → Y be a fuzzy SSPO homeomorphism from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy set A is α − SSPO compact (α∗−SSPO compact) then f(A) is an α−SSPO compact (α∗−SSPO compact). Proof. Similar to Theorem 25. Corollary 9. Let the mapping f : X → Y be a fuzzy SSPO homeomorphism from the fuzzy topological space X to a fuzzy topological space Y . If X is α − SSPO compact (α∗ − SSPO compact) then Y is an α− SSPO compact (α∗ − SSPO compact). Theorem 27. Let the mapping f : X → Y be a fuzzy SSPO homeomorphism from the fuzzy topological space X to a fuzzy topological space Y . If X is α − SSPO compact (α∗ − SSPO compact) then Y is an α-compact (α∗-compact). Proof. It follows immediately from Theorem 25. Corollary 10. Let the mapping f : X → Y be a fuzzy SSPO homeomorphism from the fuzzy topological space X to a fuzzy topological space Y . If Y is α − SSPO compact (α∗ − SSPO compact) then X is an α-compact (α∗-compact). Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 656 Proof. It follows immediately from Theorem 27. and from the fact that f−1 : Y → X is also a fuzzy SSPO homeomorphism. Definition 24. The family B of fuzzy strongly semi pre-open sets of the fuzzy topological space (X, τ) is called a base of fuzzy strongly semi pre-open sets in (X, τ) if every fuzzy strongly semi pre-open set of (X, τ) can be written as union of members of B. Theorem 28. If the fuzzy topological space (X, τ) has a countable base of fuzzy strongly semi pre-open sets then any fuzzy set A in (X, τ) is α − SSPO compact (α∗ − SSPO compact) if and only if it is countable α−SSPO compact (countable α∗−SSPO compact). Proof. It is certain that every α − SSPO compact set in the fuzzy topological space (X, τ) is also a countable α− SSPO compact fuzzy set. Conversely, let us suppose that the fuzzy set A in (X, τ) is countable α−SSPO compact. Let us suppose that the family of fuzzy strongly semi pre-open sets U = {Ui, i ∈ I} is an α − SSPO shading of A. Since (X, τ) has a countable base B = {Wi, i ∈ N} of fuzzy strongly semi pre-open sets Wi, then any fuzzy strongly semi pre-open set can be represented as union of sets from B. Now let a ∈ suppA, there exists Uj ∈ U , for some j ∈ I, such that Uj(a) > α. There are fuzzy strongly semi pre-open sets Wik), k = 1, 2, . . . ,m (note that m must not be a finite number) from B such that Uj = ∨m k=1Wik . The fact that Uj(a) > α implies the existence of Wis , is ∈ {1, 2, . . . ,m} such that Wis(a) > α. We can now claim that the family of fuzzy sets B0 = {Wik , k = 1, 2, . . . ,m} is a countable α−SSPO shading of A in (X, τ). Since, from our assumption, A is countable α−SSPO compact, there exist a finite α − SSPO subshading B1 ≤ B0. If we consider the finite collection of fuzzy strongly semi pre-open sets: U0 = {Ui : Wis ≤ Ui,Wis ∈ B0} It is obvious that U0 is a finite α − SSPO subshading of U and as consequence A is α− SSPO compact. In similar way we can prove the case when A is α∗ − SSPO compact. Theorem 29. Let the mapping f : X → Y be a fuzzy strong semi pre-continuous and surjective fuzzy SSPO-irresolute open mapping from the fuzzy topological space X to a fuzzy topological space Y . If the space X has a countable base consisting of fuzzy strongly semi pre-open sets then Y also has a countable base consisting of fuzzy strongly semi pre-open sets. Proof. Let us suppose that B = {Bi, i ∈ N} is a base for fuzzy strongly semi pre- open sets of X. Based on the assumption of the theorem it follows that f(Bi),∀i ∈ N, are fuzzy strongly semi pre-open sets in Y . If we consider the collection of fuzzy sets M = {f(Bi), i ∈ N}, and given any fuzzy strongly semi pre-open set W in Y , then again due to the conditions of the theorem, f−1(W ) is a fuzzy strongly semi pre-open set in X and it can be written in the following manner f−1(W ) = ∨i∈NBi. From the fact that mapping f is surjective, we get W = f(f−1(W )) = f(∨i∈NBi) = ∨i∈Nf(Bi). Therefore M is a base of fuzzy strongly semi pre-open sets in Y . Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 657 Definition 25. Fuzzy topological space (X, τ) is fuzzy SSPO-separable if and only if there exists a countable sequence of fuzzy points {pi}i∈N such that for each U ∈ FSSPO(τ), U ̸= 0X , there exists a fuzzy point pj such that pj ∈ U , for some j ∈ N. It is obvious that the concept of fuzzy SSPO-separability is the generalization of fuzzy separability. If a fuzzy topological space is fuzzy SSPO-separable then it is also fuzzy separable. Theorem 30. If the fuzzy topological space (X, τ) has a countable base B of fuzzy strongly semi pre-open sets then (X, τ) is an SSPO-separable space. Proof. Let us suppose that B = {Bi, i ∈ N} is a countable base for fuzzy strongly semi pre-open sets in (X, τ). Let us consider any member of B, let it be denoted as Bj , such that Bj ̸= 0X , then there exists a fuzzy point xj ∈ X such that Bj(xj) > 0. If we now define a fuzzy point as follows:{ pj(x) = Bj(xj) if x = xj pj(x) = 0 if x ̸= xj We can conclude that pj ≤ Bj . Let us consider the corresponding countable sequence of fuzzy points {pi}i∈N. Given any fuzzy strongly semi pre-open set U in (X, τ), it must contain a certain Bs ∈ B and therefore there exists a fuzzy point ps ≤ Bs such that ps ≤ U . In other words X is SSPO-separable space. The converse of this theorem does not stand. Let X be an infinite set, and let fts(X, τ) be such that any fuzzy open set in τ contains a fuzzy singleton p ∈ X (or a countable set of singletons). In this case (X, τ) does not contain a fuzzy countable base consisting of fuzzy open sets and it does not contain a countable base of fuzzy strongly semi pre-open sets. Theorem 31. Let the mapping f : X → Y be a fuzzy SSPO-irresolute and surjective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the space X is fuzzy SSPO-separable then Y is a fuzzy SSPO-separable space. Proof. Let us consider a countable sequence of fuzzy points {pi}i∈N from X such that for any fuzzy strongly semi pre-open set W in X, W ̸= 0X , there exists a fuzzy point pi such that pi ≤ W . The sequence {f(pi)}i∈N is a countable sequence of fuzzy points in Y . Let us suppose that V is a fuzzy strongly semi pre-open set in Y such that V ̸= 0Y . From the assumption of the theorem f−1(V ) is a fuzzy strongly semi pre-open set in X and f−1(V ) ̸= 0X . Because the space X is fuzzy SSPO-separable then there exists a fuzzy point ps such that ps ≤ f−1(V ). Now f(ps) ≤ f(f−1(V )) = V , and we have shown that {f(pi)}i∈N is a countable sequence of fuzzy points in Y such that for any fuzzy strongly semi pre-open set V in Y , V ̸= 0Y , there exists a fuzzy point f(ps) such that ps ≤ V , that is Y is a fuzzy SSPO-separable space. Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 658 Theorem 32. Let the mapping f : X → Y be a fuzzy strong semi pre-continuous and surjective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the space X is fuzzy SSPO-separable then the space Y is fuzzy separable space. Proof. Similar to Theorem 31 Theorem 33. Let the mapping f : X → Y be a fuzzy SSPO homeomorphism from the fuzzy topological space X to a fuzzy topological space Y . If the space X is SSPO-separable then Y will also be SSPO-separable space. Proof. Let us consider a countable sequence of fuzzy points {pi}i∈N from X such that for any fuzzy strongly semi pre-open set W in X, W ̸= 0X , there exists a fuzzy point pi such that pi ≤ W . Then the sequence {f(pi)}i∈N is a countable sequence of fuzzy points in Y . Let us suppose that V is a fuzzy strongly semi pre-open set in Y such that V ̸= 0Y . From the assumption of the theorem f−1(V ) is a fuzzy strongly semi pre-open set in X and f−1(V ) ̸= 0X . Due to the fact that the space X is fuzzy SSPO-separable then there exists a fuzzy point ps such that ps ≤ f−1(V ). Now f(ps) ≤ f(f−1(V )) = V and we have shown that {f(pi)}i∈N is a countable sequence satisfying the conditions of definition 25, therefore Y is a fuzzy SSPO-separable space. Definition 26. The fuzzy set A of the fuzzy topological space (X, τ) is called α− SSPO Lindelof (respectively α∗ − SSPO Lindelof) if every α − SSPO shading (α∗ − SSPO shading) of the set A has a countable α− SSPO subshading (countable α∗ − SSPO sub- shading). If instead of the fuzzy set A we consider space X then we can state that the fuzzy topological space X is α− SSPO Lindelof (respectively α∗ − SSPO Lindelof). It is certain that from the above definition we can conclude that every α − SSPO compact (α∗−SSPO compact) space is also an α−SSPO Lindelof (α∗−SSPO Lindelof) space. Every α − SSPO Lindelof (α∗ − SSPO Lindelof) space is an α- Lindelof (α∗-Lindelof) space. Theorem 34. If the fuzzy topological space (X, τ) has a countable base B of fuzzy strongly semi pre-open sets then (X, τ) is an α−SSPO Lindelof (respectively α∗−SSPO Lindelof) space. Proof. Similar to Theorem 28. Theorem 35. Let the fuzzy set A of the fuzzy topological space (X, τ) be an α − SSPO Lindelof (α∗ − SSPO Lindelof) set. The fuzzy set A is countable α − SSPO compact (countable α∗ − SSPO compact) if and only if A is α − SSPO compact (α∗ − SSPO compact). Sh. Makolli, B. Krsteska / Eur. J. Pure Appl. Math, 17 (2) (2024), 638-662 659 Proof. It is certain that every α − SSPO compact set in the fuzzy topological space (X, τ) is also a countableα− SSPO compact fuzzy set. Conversely, let us suppose that the fuzzy set A in (X, τ) is a countable α−SSPO compact set. Let us suppose that the family of fuzzy strongly semi pre-open sets U = {Ui, i ∈ I} is an α − SSPO shading of A. Since the fuzzy set A of (X, τ) is an α − SSPO Lindelof then there exists a countable α − SSPO subshading V1 of α − SSPO shading U . Based on the assumption that the fuzzy set A is countable α − SSPO compact set, then there exists a finite α−SSPO subshading V2 of α−SSPO shading U . It is clear that given an α− SSPO shading U of the fuzzy set A there exists a finite α− SSPO subshading V2 of U and subsequently the fuzzy set A is α− SSPO compact. In similar way we can prove the case when A is an α∗ − SSPO Lindelof set. Theorem 36. Let A be an α−SSPO Lindelof (α∗−SSPO Lindelof) fuzzy set in (X, τ) and let B ∈ FSSPC(τ), then A ∧ B is an α − SSPO Lindelof (α∗ − SSPO Lindelof) fuzzy set in the fuzzy topological space (X, τ). Proof. Similar to Theorem 22. Corollary 11. Let X be an α− SSPO Lindelof (α∗ − SSPO Lindelof) space, then any fuzzy set B ∈ FSSPC(τ) is an α − SSPO Lindelof (α∗ − SSPO Lindelof) fuzzy set in the fuzzy topological space (X, τ). Proof. It follows directly from Theorem 36. Theorem 37. Let A,B be α−SSPO Lindelof (α∗−SSPO Lindelof) fuzzy sets in (X, τ), then the fuzzy set A ∨ B is also an α − SSPO Lindelof (α∗ − SSPO Lindelof) fuzzy set in (X, τ). Proof. In similar way as Theorem 23. Theorem 38. If f : X → Y is a fuzzy SSPO-irresolute mapping from the fuzzy topological space X to a fuzzy topological space Y . If the fuzzy set A is an α − SSPO Lindelof (α∗ − SSPO Lindelof) fuzzy set in X then f(A) is an α− SSPO Lindelof (α∗ − SSPO Lindelof) fuzzy set in Y . Proof. In a similar way as Theorem 24. Corollary 12. If f : X → Y is a fuzzy SSPO-irresolute mapping from the fuzzy topolog- ical space X to a fuzzy topological space Y . If X is an α − SSPO Lindelof (α∗ − SSPO Lindelof) then f(X) is an α− SSPO Lindelof (α∗ − SSPO Lindelof) fuzzy set in Y . Proof. It follows directly from Theorem 38. Corollary 13. Let f : X → Y is a fuzzy SSPO-irresolute and surjective mapping from the fuzzy topological space X to a fuzzy topological space Y . If X is an α−SSPO Lindelof (α∗ − SSPO Lindelof) then Y is an α− SSPO Lindelof (α∗ − SSPO Lindelof). REFERENCES 660 Proof. It follows from Theorem 38. We can also show that the following assertion are true. Theorem 39. Let the mapping f : X → Y be a fuzzy strong semi pre-continuous and surjective mapping from the fuzzy topological space X to a fuzzy topological space Y . If the space X is an α − SSPO Lindelof (α∗ − SSPO Lindelof) then Y will be an α- Lindelof (α∗- Lindelof) space. Theorem 40. Let the mapping f : X → Y be a fuzzy SSPO homeomorphism from the fuzzy topological space X to a fuzzy topological space Y and let A be a fuzzy set in X. If the set A is an α − SSPO Lindelof (α∗ − SSPO Lindelof) set in X then f(A) will also be an α− SSPO Lindelof (α∗ − SSPO Lindelof) fuzzy set in Y . Corollary 14. Let the mapping f : X → Y be a fuzzy SSPO homeomorphism from the fuzzy topological space X to a fuzzy topological spaceY . If X is an α − SSPO Lindelof (α∗−SSPO Lindelof) then Y will also be an α−SSPO Lindelof (α∗−SSPO Lindelof). 6. Conclusion In this paper we have investigated properties of a new form of fuzzy pre-separation axioms as well as the new form of fuzzy compactness induced by the new class of fuzzy generalized opened sets. We also investigated their properties in regards to the fuzzy strong semi pre-continuous functions as well as the fuzzy SSPO-irresolute mappings. We have shown that the concept of fuzzy strong pre-separation axioms is stronger than the ordinary fuzzy separation axioms. From the properties that we investigated, the concept of α−SSPO Lindelof space, fuzzy SSPO-separability and the existence of a base consisting of fuzzy strongly semi pre-open sets, the strongest concept appears to be the concept of the existence of a base consisting of fuzzy strongly semi pre-open sets. Our future work will be focused on introducing new form of fuzzy connectedness which will be stronger than the concepts of fuzzy connectedness introduced by other authors. 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