extension of generalized topological spaces via stacks - Springer Link

Acta Math. Hungar., 147 (1) (2015), 1–11 Acta Math. Hungar. DOI: 10.1007/s10474-015-0537-2 DOI: 0 First published online July 30, 2015

EXTENSION OF GENERALIZED TOPOLOGICAL SPACES VIA STACKS A. DEB RAY Department of Mathematics, West Bengal State University, 700126 Berunanpukuria, P.O. Malikapur, North 24 Parganas, India e-mail: [email protected] (Received September 5, 2013; revised April 22, 2015; accepted May 18, 2015)

Abstract. A typical construction for extensions of T0 generalized topological spaces (in short, GTS) is introduced and studied. A precise description for such extensions to be µ-compact is also established. The construction of compactification of spaces in terms of grills follow as a special case of the results obtained in this paper, as µ-compactness of a GTS is a generalization of compactness of topological spaces.

1. Introduction and preliminaries Generalized topological space (GTS) under the name supratopological space, was introduced by A. S. Mashhour, A. A. Allam, F. S. Mahmoud, F. H. Khedr [11], where X was required to be generalized open. However, generalized topological spaces have been studied extensively by vari´ Császár [3]. It ous mathematicians after its systematic development, by A. has been observed that the collection of all open, δ-open [18], θ-open [18], semiopen [10], preopen [12] and several other collection of sets on a topological space form generalized topologies on X. Therefore, one may utilize the concept of generalized topology to unify certain existing theories developed in terms of open sets, δ-open sets, θ-open sets, semiopen sets, preopen sets, etc. In [2], we have unified the concept of cluster-like sets, viz., cluster sets [7], δ-cluster sets [14], θ-cluster sets [9], S-cluster sets [13], s-cluster sets [1], p-cluster sets [15] and many more, by giving a common generalization of all these [2]. Compactifications of topological spaces have been studied widely by various mathematicians of different times. It was observed by K. C. Chattopadhyay and W. J. Thron [5] that from any T0 topological space (X, τ ) one may Key words and phrases: fp-stack, c-stack, extension of GTS, principal extension, µ-compact extension, generalized linkage compact. Mathematics Subject Classification: 54A05, 54D20. c 2015 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary

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A. A. DEB DEB RAY RAY

construct X ∗ , a collection of grills on X and topologize X ∗ in such a way that X is embedded in X ∗ as a dense subspace. In this paper, we show that generalized topological spaces possess extensions via stacks, a generalization of grills. Introducing extensions of GTS we establish a set-theoretic characterization for such an extension to be µ-compact, a unification of compactness, near compactness [16], strong compactness [8], semicompactness [6], etc. Constructing µ-compact extension of a T0 GTS we see that the results for compact extensions of topological spaces follow as a special case. We now state a few definitions and results that are required in this paper. For any set X, a nonempty collection of subsets S of X is said to form a stack [17] if ∅ �∈ S and A ∈ S, B A implies B ∈ S. A stack is called a grill [17] if A ∪ B ∈ S implies A ∈ S or B ∈ S. It is wellknown that a stack is called a filter if A, B ∈ S implies A ∩ B ∈ S and a filter is an ultrafilter if for any A X, either A or X \ A ∈ S. G is a grill if and only if {X \ A : A �∈ G} is a filter. (The same formula defines a filter by the concept of a grill.) Let X be a nonempty set and µ be a collection of subsets of X (i.e. µ P(X)). µ is called a generalized topology (briefly GT) [3] on X iff ∅ ∈ µ and Gλ ∈ µ for λ ∈ Λ(�= φ) implies ∪λ∈Λ Gλ ∈ µ. The pair (X, µ) is called a generalized topological space (briefly GTS). If further, X ∈ µ then such a GT is called a strong GT and the pair (X, µ) is called a strong GTS. The elements of µ are µ-open sets and their complements are known as µ-closed sets. The set of all µ-open sets containing an element x ∈ X is denoted by µ(x). A base [4] for a GT µ on X is a collection β of subsets of X such that µ = {∪β ∗ : β ∗ β}. It has also been established in [4] that any collection of subsets of X forms a base for some GT on X. The generalized closure operator cµ : P(X) → P(X) on a GTS [19] is defined as follows: For any A X, cµ (A) = {x ∈ X : for each U ∈ µ(x), we have U ∩ A �= ∅}. It may be observed that cµ has the following properties: • A cµ (A), for any A X, • cµ (A) cµ (B) whenever A B, • cµ (cµ(A)) = cµ (A). In case µ is a strong GT then cµ (∅) = ∅ also holds. On the other hand, one may define an operator c : P(X) → P(X) on a set X such that the operator satisfies the following conditions: • c(∅) = ∅, • A c(A), for any A X, • c(A) c(B) whenever A B, • c(c(A)) = c(A). Then such an operator generates a strong GT on X given by µ = {U X : c(X \ U ) = X \ U } such that cµ (A) = c(A), for each A X. The dual operator iµ may also be defined as iµ (A) = X \ cµ (X \ A), for each A X. The µ-neighbourhood system η(x) = {A X : for some U ∈ µ, we have x ∈ U A} forms a stack on X. A GTS (X, µ) is said to be Acta Hungarica 147, 2015 Acta Mathematica Mathematica Hungarica

EXTENSION OF GENERALIZED GENERALIZED TOPOLOGICAL TOPOLOGICAL SPACES SPACES VIA VIA STACKS STACKS EXTENSION OF

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• T0 [19], if for any x1 �= x2 in X, there exists U ∈ µ, such that x1 ∈ U and x2 �∈ U or x1 �∈ U and x2 ∈ U ; • µ-regular [19], if for any µ-closed set A and x �∈ A, there exists U ∈ µ, such that x ∈ U cµ (U ) X \ A. 2. Stacks and extensions of GTS In this section, we introduce various stacks formed on a GTS depending on the generalized topological structure of the space. Further, a construction of an extension of a GTS is obtained via a suitable collection of stacks on the space. Let (X, µ) be a GTS and cµ be the µ-closure operator of the given GTS. It is easy to see that x ∈ cµ (∅) if and only if there does not exist any U ∈ µ such that x ∈ U . So, we have the following definition: Definition 2.1. In a GTS (X, µ), if cµ (∅) �= ∅, then each point of cµ (∅) is called an abandoned point of X. It is to be noted that a GTS is strong if and only if it has no abandoned point. The following theorem shows that for every non-abandoned point x ∈ X, one can always get a stack on the generalized topological space (X, µ). Definition 2.2. For each nonabandoned x ∈ X, by S(x) we denote the collection {A X : x ∈ cµ (A)}. Theorem 2.1. For each nonabandoned x ∈ X , S(x) is a stack on X . Proof. ∅ �∈ S(x) ∋ X, hence S(x) �= ∅. Also, A ∈ S(x) and A B implies x ∈ cµ (A) cµ (B). Hence, B ∈ S(x). Corollary 2.1. In a strong GTS (X, µ), for every x ∈ X , S(x) = {A X : x ∈ cµ (A)} is a stack. Remark 2.1. If in particular X is a topological space then each S(x) induces stronger properties and becomes a grill on X called an adherence grill [17] on X. We name a few stacks which play significant role in our discussion: Definition 2.3. A stack S on a GTS is called a (i) closure stack (in short, c-stack), if cµ (A) ∈ S ⇒ A ∈ S, for each A X; (ii) fixed point stack (in short, fp-stack), if S = S(x) = {A X : x ∈ cµ (A)}, for some x ∈ X; (iii) neighbourhood stack (in short, nbd -stack), if S = µ-neighbourhood system at x, for some x ∈ X. Acta Mathematica Hungarica Hungarica 147, 2015 Acta Mathematica

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Remark 2.2. It is easy to observe that an fp-stack is always a c-stack. However, there exist several examples of c-stacks other than fp-stacks, e.g., the stack consisting of all dense subsets of a T2 topological space X with |X| ≥ 2. Definition 2.4. If S is any stack on a GTS (X, µ), then dual(S) = {A X : X \ A �∈ S} is called the dual of S. Theorem 2.2. For any stack S on a GTS (X, µ), dual(S) is a stack on X . Proof. Straightforward. One can easily observe the following: • dual(S) = {A X : ∀ B ∈ S we have B ∩ A �= ∅}. • If S and T are two stacks with S T , then dual(T ) dual(S). • For any stack S, dual(dual(S)) = S. • If U is an ultrafilter on a GTS (X, µ), then dual(U) = U . Theorem 2.3. dual(S(x)) = η(x), the neighbourhood stack at x. Proof. For each x ∈ X, A ∈ dual(S(x)) ⇔ X \ A �∈ S(x) ⇔ x �∈ cµ (X \ A) ⇔ x ∈ iµ (A).

Corollary 2.2. dual(η(x)) = S(x). Proof. Follows from Theorem 2.3 and the fact that dual(dual(S(x))) = S(x). Remark 2.3. The dual of an fp-stack is not necessarily a c-stack. Definition 2.5. A GTS (Y, σ) is called an extension of a GTS (X, µ), if there is an injective map ψ : X → Y such that ψ(cµ (A)) = cσ (ψ(A)) ∩ ψ(X), for each A X and cσ (ψ(X)) = Y . Definition 2.6. Two extensions (Y1 , σ1 ) and (Y2 , σ2 ) of a GTS (X, µ) via the injective maps ψ and η respectively are said to be equivalent if there exists a generalized homeomorphism χ : Y1 → Y2 , such that χ ◦ ψ = η. Definition 2.7. If (Y, σ) is an extension of a GTS (X, µ) via the injective map ψ : X → Y , then for each y ∈ Y , T (y) = {A X : y ∈ cσ (ψ(A))} is called the trace of y on X. Theorem 2.4. If (Y, σ) is an extension of a GTS (X, µ) via the injective map ψ : X → Y , then for each y ∈ Y , the trace of y on X (i.e., T (y) = {A X : y ∈ cσ (ψ(A))}) is a c-stack on X . Acta Hungarica 147, 2015 Acta Mathematica Mathematica Hungarica


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Proof. Clearly ∅ �∈ T (y). If A ∈ T (y) and B A then y ∈ cσ (ψ(A)) cσ (ψ(B)) and therefore B ∈ T (y). Moreover, X ∈ T (y), since y ∈ cσ (ψ(X)) = Y . Hence, T (y) is a stack on X. If cµ (A) ∈ T (y), then y ∈ cσ (ψ(cµ (A))) = cσ (cσ (ψ(A)) ∩ ψ(X) cσ (cσ (ψ(A)) = cσ (ψ(A)). Hence, A ∈ T (y), proving that T (y) is a c-stack on X. Remark 2.4. If X is a topological space and Y an extension of X then each T (y) is a c-grill on X. The construction in the next theorem is a generalization of the strict extension for a collection of filters of closed sets on a topological space, if we rewrite the definition in terms of the duals of the c-stacks. Observe that with Ac as defined in the next theorem, X ∗ \ Ac = {S ∈ X ∗ : X \ A ∈ dual(S)}. Theorem 2.5. Let (X, µ) be a GTS. Consider any collection X ∗ of c-stacks on X . For each A X , define Ac = {S ∈ X ∗ : A ∈ S}. Then {Ac : σ on X ∗ . A X} forms a base for generalized closed sets for some cstrong GT ∗ ∗ c Moreover, cσ : P(X ) → P(X ) is given by cσ (α) = {A : α A , A X}, for each α X ∗ . Proof. Consider B = {Ac : A X}. For B to form a base for generalized closed sets on X ∗ for some strong GT, we need to check whether ∅, X ∗ are in B. But that is clear as ∅c = ∅ and X c = X ∗ . The rest of the assertion is immediate. Theorem 2.6. If (Y, σ) is an extension of a GTS (X, µ) via the injective map ψ , then for each x ∈ X , T (ψ(x)) = S(x), the fp-stack at x. Proof. Follows from the definition of an extension and injectiveness of the map ψ. In what follows, by a GTS we always mean a GTS without any abandoned point, i.e., a strong GT. Theorem 2.7. Let (X, µ) be a T0 GTS and X ∗ be the collection of all c-stacks on X , with the strong GT σ given bythe generalized closure operator cσ : P(X ∗) → P(X ∗), described as cσ (α) = {Ac : α Ac , A X}. Then (X ∗ , σ) is an extension of (X, µ) via the map ψ : (X, µ) → (X ∗ , σ) given by ψ(x) = S(x). Proof. Define a map ψ : X → X ∗ by ψ(x) = S(x). Since (X, µ) is T0 , for any x1 �= x2 , there exists U ∈ µ such that x1 ∈ U and x2 �∈ U or x1 �∈ U and x2 ∈ U . Without loss of generality, we assume x1 ∈ U and x2 �∈ U . So, x2 ∈ X \ U = cµ (X \ U ). Consequently, X \ U ∈ S(x2 ). But x1 ∈ U ⇒ x1 �∈ X \ U = cµ (X \ U ) and therefore, X \ U �∈ S(x1 ). Hence, S(x1 ) �= S(x2 ), proving the map ψ to be injective. Acta Mathematica Hungarica Hungarica 147, 2015 Acta Mathematica

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To show that ψ(cµ (A)) = cσ (ψ(A)) ∩ ψ(X), for any A X. We prove this in two steps as follows: (i) ψ(cµ (A)) = Ac ∩ ψ(X), for any A X. (ii) cσ (ψ(A)) = Ac , for any A X. Again, in order to show (ii), we need to verify that (1)

for any A X, ψ(A) Ac

and (2)

whenever ψ(A) B c , we have that Ac B c .

To show (i), observe x ∈ cµ (A) ⇔ A ∈ S(x) = ψ(x) ⇔ ψ(x) ∈ Ac ∩ ψ(X). To prove (1), we observe x ∈ A ⇒ x ∈ cµ (A) ⇔ A ∈ S(x) = ψ(x) ⇒ ψ(x) ∈ Ac . Hence, ψ(A) Ac . To prove (2), let ψ(A) B c , for some B X. Then x ∈ A ⇒ ψ(x) ∈ ψ(A) B c and so, B ∈ ψ(x) = S(x), i.e., x ∈ cµ (B) which means A cµ (B). Now, S ∈ Ac ⇔ A ∈ S and A cµ (B) implies cµ (B) ∈ S and therefore, S ∈ (cµ (B))c . But (cµ (B))c = B c , by the definition of a c-stack. Hence, Ac B c , as desired. Now in particular, putting A = X in equation (ii), cσ (ψ(X)) = X c = X ∗ . So, (X ∗, σ) becomes an extension of the GTS (X, µ). Theorem 2.8. Let (X, µ) be a T0 GTS and Y any collection of c-stacks on X containing all fp-stacks on X . Then Y as a subspace of X ∗ (with X ∗ constructed in Theorem 2.7) also forms an extension of (X, µ). Proof. The proof follows from the proof of Theorem 2.7. Theorem 2.9. Let (X, µ) be a T0 GTS and (X ∗ , σ) be the extension of (X, µ) from Theorem 2.7 via the map ψ : (X, µ) → (X ∗ , σ) given by ψ(x) = S(x). Then (X ∗, σ) has the following two properties: (i) For all S1 , S2 ∈ X ∗ , with S1 �= S2 , we have T (S1 ) �= T (S2 ). (ii) The collection {cσ (ψ(A)) : A X} constitutes a base for generalized closed sets of σ . Proof. In view of Theorem(2.7), cσ (ψ(A)) = Ac . So, T (S1 ) = {A X : S1 ∈ cσ (ψ(A))} = {A X : S1 ∈ Ac } = {A X : A ∈ S1 } = S1 �= S2 = T (S2 ) Also, from cσ (ψ(A)) = Ac and Theorem 2.7, it follows immediately that {cσ (ψ(A)) : A X} forms a base for generalized closed sets of (X ∗ , σ). Acta Hungarica 147, 2015 Acta Mathematica Mathematica Hungarica


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Definition 2.8. An extension (Y, σ) of a GTS (X, µ) via the map ψ : (X, µ) → (Y, σ), is said to be a principal extension if the following two conditions are satisfied: (i) For all y1 �= y2 ∈ Y , T (y1 ) �= T (y2 ). (ii) The collection {cσ (ψ(A)) : A X} constitutes a base for generalized closed sets of σ. In view of Theorem (2.9) it is evident that the extension obtained in Theorem (2.7) is a principal extension of a T0 GTS (X, µ). Theorem 2.10. Any two principal extensions of a T0 GTS (X, µ) which have identical family of traces, are equivalent. Proof. Suppose (Y1 , σ1 ) and (Y2 , σ2 ) are two principal extensions of a GTS (X, µ), via the maps ψ1 and ψ2 . By the given condition {T (y1 ) : y1 ∈ Y1 } = {T (y2) : y2 ∈ Y2 }. Define a map f : Y1 → Y2 by f (y1) = y2 whenever T (y1 ) = T (y2 ). This map f is well defined and injective as both Y1 and Y2 are principal extensions. Then t ∈ cσ1 (ψ1 (A)) ⇔ A ∈ T (t); T (t) = T (s) for some s ∈ Y2 implies that A ∈ T (s) and f (t) = s. But A ∈ T (s) ⇔ s ∈ cσ2 (ψ2 (A)). Consequently, f (cσ1 (ψ1 (A))) cσ2 (ψ2 (A)). The reverse implication follows similarly. Since {cσ1 (ψ1 (A)) : A X} and {cσ2 (ψ2 (A)) : A X} are bases for generalized closed sets of the GTS (Y1 , σ1 ) and (Y2 , σ2 ) respectively, the equivalence follows. Also, for any x ∈ X, as T (ψ1 (x)) = S(x) = T (ψ2 (x)), f (ψ1 (x)) = ψ2 (x). Corollary 2.3. Any principal extension of a T0 GTS (X, µ) is equivalent to a subspace of the extension (X ∗ , σ) (as described in Theorem 2.7). Proof. Suppose (Y, δ) is a principal extension of (X, µ) via the injective map η. Then each T (y) is a c-stack on X. So, T (y) ∈ X ∗ , for each y ∈ Y . Also, T (S) = S, for each S ∈ X ∗ . i.e., T (T (y)) = T (y). Consider Z ∗ = {T (y) : y ∈ Y } X ∗ . The trace of Z ∗ is the collection {T (S) : S ∈ Z ∗ } = {T (T (y)) : y ∈ Y } = {T (y) : y ∈ Y }, which is the trace of Y . Hence by the Theorem 2.10, Z ∗ with respect to the subspace generalized topology induced from (X ∗ , σ) is equivalent to the given principal extension of X. 3. µ-compact extensions We begin with definitions that are required in this section. Definition 3.1. A GTS (X, µ) is said to be µ-compact, if any µ-open cover of X has a finite subcover. Acta Mathematica Hungarica Hungarica 147, 2015 Acta Mathematica

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If X is a topological space, then for different choices of the GT µ on X one gets various known compact-like spaces. A few of these are listed in the following table: µ µ-compact open sets compact δ-open sets near compact [16] semi-open sets semi-compact [6] pre-open sets strong compact [8] Definition 3.2. A stack (or, grill) S is said to µ-adhere at some x in a GTS (X, µ), if S S(x). Theorem 3.1. For a GTS (X, µ), the following are equivalent. (i) (X, µ) is µ-compact. (ii) Every ultrafilter on X µ-adheres. (iii) Every filter on X µ-adheres. (iv) For any collection of µ-closed sets{Fα } with ∩Fα = ∅ there exist finitely many Fα1 , Fα2 , . . . , Fαn such that ni=1 Fαi = ∅. Proof. The proof is analogous to the proof for topological spaces.

Definition 3.3. A stack (or, grill) S on a GTS (X, µ) is called (i) linked if for any pair A, B ∈ S, cµ (A) ∩ cµ (B) �= ∅; (ii) conjoint if for any A1 , . . . , An ∈ S, ∩ni=1 cµ (Ai ) �= ∅. Certainly, it follows from the definition that a conjoint stack (or grill) is always linked. We show how µ-adherence of conjoint stacks (or conjoint grills) characterize µ-compactness of a GTS. Theorem 3.2. For a GTS (X, µ), the following are equivalent. (i) (X, µ) is µ-compact. (ii) Every conjoint stack µ-adheres in X . (iii) Every conjoint grill µ-adheres in X . Proof. (i) ⇒ (ii). Suppose, (X, µ) is µ-compact. Let Ω be any conjoint stack on X. Then {cµ (A) : A ∈ Ω} is a collection of µ-closed sets in (Ai ) �= ∅, for any n and any A1 , A2 , . . . , An ∈ Ω. So, by X with ni=1 cµ µ-compactness, A∈Ω cµ (A) �= ∅. Let x ∈ A∈Ω cµ (A). Then A ∈ S(x), for every A ∈ Ω, i.e., Ω µ-adheres in X. (ii) ⇒ (iii). Follows from the fact that a conjoint grill is a conjoint stack. (iii) ⇒ (i). Suppose U is an ultrafilter on X. Since U is a conjoint grill, U µ-adheres in X. Definition 3.4. A GTS (X, µ) is called generalized linkage compact, if every linked grill µ-adheres in X. Acta Hungarica 147, 2015 Acta Mathematica Mathematica Hungarica


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By Theorem 3.2, a generalized linkage compact GTS is certainly µcompact. The next theorem establishes that in a µ-regular GTS these two concepts are equivalent. Theorem 3.3. A µ-regular µ-compact GTS is generalized linkage compact. Proof. Let Ω be a linked grill on a µ-regular GTS (X, µ) which does not µ-adhere in X. For each x ∈ X, there exists Ax ∈ Ω such that Ax �∈ S(x). So, x �∈ cµ (Ax ). By µ-regularity of X, there exists Vx ∈ µ containing x such that cµ (Vx ) ∩ cµ (Ax ) = ∅. Since Ω is linked and Ax ∈ Ω, it follows that Vx �∈ Ω. On the other hand, {Vx : x ∈ X} constitutes a µ-cover ofX. By n µ-compactness of X, there exist Vx1 , V x2 , . . . , Vxn such that X = i=1 Vxi . n As Ω is a grill on X, X ∈ Ω as well as i=1 Vxi �∈ Ω, a contradiction. Hence the GTS is generalized linkage compact. We conclude this section by obtaining a precise set-theoretic description of the extension (X ∗ , σ) (as constructed in Theorem 2.7) of a GTS (X, µ) to be µ-compact (respectively, generalized linkage compact). Definition 3.5. Let ∆ = {Sα : α ∈ Λ} be a collection of stacks on a GTS (X, µ). ∆ is called a finitely determined collection (respectively, binary collection), if whenever any finitely many members (respectively, a pair of members) of a grill G on X are contained in some member of ∆, then there exists some S0 ∈ ∆ such that G S0 . Theorem 3.4. Let (X ∗, σ) be the extension (as constructed in Theorem 2.7) of a T0 GTS (X, µ) via the injective map φ : X → X ∗ . (X ∗ , σ) is µ-compact if and only if X ∗ (which contains all fp-stacks) is a finitely determined collection of c-stacks on X . Proof. Suppose (X ∗ , σ) is µ-compact. The very construction of X ∗ implies that it is a collection of c-stacks containing all fp-stacks on X. We need only to show that it is a finitely determined collection of stacks on X. Let G be a grill on X such that for any finite family A1 , . . . , An ∈ G, there is an S ∈ X ∗ such that A1 , . . . , An ∈ S. To show that G S0 , for some S0 ∈ X ∗ , observe A1 , . . . , An ∈ S ⇒ S ∈ Ac1 ∩ Ac2 ∩ · · · ∩ Acn , so that Ac1 ∩ Ac2 ∩ · · · ∩ Acn �= ∅. Define Ω = {α ⊂ X ∗ : φ(A) α for some A ∈ G}. Then Ω is a conjoint stack on X ∗ : ∅ �∈ Ω as ∅ �∈ G; α ∈ Ω and α β ⇒ φ(A) α β, for some A ∈ G and so, β ∈ Ω and X ∗ ∈ Ω. For any finite family α1 , . . . , αn ∈ Ω, there are A1 , . . . , An ∈ G such that φ(Ai ) αi , i = 1, 2, . . . , n. Consequently, n

i=1

cσ (αi )

n

i=1

cσ (φ(Ai )) =

n

Aci �= ∅,

i=1

Acta Mathematica Hungarica Hungarica 147, 2015 Acta Mathematica

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proving that Ω is a conjoint stack. Using Theorem 3.2, Ω µ-adheres in X ∗ , say at S0 . i.e., Ω S(S0 ). To show that G S0 , let A ∈ G. Then φ(A) ∈ Ω and so S0 ∈ cσ (φ(A)) = Ac . Therefore, A ∈ S0 , as desired. Conversely, let X ∗ be a finitely determined collection of c-stacks (that contains all fp-stacks on X). To show that (X ∗ , σ) is µ-compact we apply again Theorem 3.2. Let G be a conjoint grill on X ∗ . Define A = {A X : φ(A) ∈ G} and B = {A X : Ac \ φ(X) ∈ G}. Then ∅ �∈ A, B and B ⊃ A ∈ A ⇒ B ∈ A,

B ⊃ A ∈ B ⇒ B ∈ B,

A ∪ B ∈ A ⇒ (A ∈ A ∨ B ∈ A) and A ∪ B ∈ B ⇒ (A ∈ B ∨ B ∈ B). Since X ∗ ∈ G, therefore either φ(X) ∈ G, and then X ∈ A, or X c \ φ(X) = X ∗ \ φ(X) ∈ G, and then X ∈ B. So, A ∪ B is a grill on X. Let G1 , . . . , Gn ∈ A ∪ B. Without loss of generality, we assume that G1 , . . . , Gk ∈ A and Gk+1 , . . . , Gn ∈ B. Then φ(Gi ) ∈ G and Gcj \ φ(X) ∈ G, i = 1, 2, . . . , k and j = k + 1, . . . , n. Since G is a conjoint grill, k

i=1

cσ (φ(Gi )) ∩

n

j=k+1

cσ (Gcj

\ φ(X))

�= ∅.

Therefore, ( ki=1 Gci ) ∩ ( nj=k+1 cσ (Gcj )) �= ∅ which implies that ni=1 Gci �= ∅. Hence, there exists some S ∈ X ∗ such that G1 , G2 , . . . , Gn ∈ S. As X ∗ is a finitely determined collection of stacks, there exists some S0 ∈ X ∗ such that A ∪ B S0 . We now show that G S(S0 ), showing compactness of (X, µ), by Theorem 3.2. If α ∈ G then α X ∗ . Again α = (α ∩ φ(X)) ∪ (α \ φ(X)) and hence, either α ∩ φ(X) ∈ G or α \ φ(X) ∈ G. If α ∩ φ(X) ∈ G, then φ−1 (α) ∈ A. So, φ−1 (α) ∈ S0 , therefore S0 ∈ (φ−1 (α))c = cσ (φ(φ−1 )(α)) cσ (α), whence α ∈ S(S0 ). If α \ φ(X) ∈ G, then assume that Ac is a basic σ-closed set in X ∗ such that α Ac . Then α \ φ(X) Ac \ φ(X) and so, Ac \ φ(X) ∈ G. S0 ∈ Ac for any basic σ-closed Hence A ∈ B. Consequently, A c∈ S0 . Thus, set containing α, i.e., S0 ∈ {A : α ⊂ Ac } = cσ (α) so that α S(S0 ). Theorem 3.5. Let (X ∗, σ) be the extension (as constructed in Theorem 2.7) of a T0 GTS (X, µ) via the injective map φ : X → X ∗ . (X ∗, σ) is generalized linkage compact if and only if X ∗ (that contains all fp-stacks) is a binary collection of c-stacks on X . Proof. Similar to the proof of Theorem 3.4 and hence omitted. Acknowledgement. The author is grateful to the referee for critical comments and valuable suggestions towards improvement of the paper. Acta Hungarica 147, 2015 Acta Mathematica Mathematica Hungarica


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