EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5362 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Category of (i, j)-Baire Bilocales Mbekezeli Nxumalo Department of Mathematics (Pure and Applied), Faculty of Science, Rhodes University, Makhanda, Eastern Cape, South Africa Abstract. We define and characterize the notion of (i, j)-Baireness for bilocales. We also give in- ternal properties of (i, j)-Baire bilocales which are not translated from properties of (i, j)-Baireness in bispaces. It turns out (i, j)-Baire bilocales are conservative in bilocales, in the sense that a bitopological space is almost (i, j)-Baire if and only if the bilocale it induces is (i, j)-Baire. Fur- thermore, in the class of Noetherian bilocales, (i, j)-Baireness of a bilocale coincides with (i, j)- Baireness of its ideal bilocale. We also consider relative versions of (i, j)-Baire where we show that a bilocale is (i, j)-Baire only if the subbilocale induced by the Booleanization is (i, j)-Baire. We use the characterization of (i, j)-Baire bilocales to introduce and characterize (τi, τj)-Baireness in the category of topobilocales. 2020 Mathematics Subject Classifications: 06D22, 54E52, 54E55 Key Words and Phrases: (i, j)-Baire, topobilocale, i-prefit, i-pseudocomplete, ideal bilocale, relatively (i, j)-Baire 1. Introduction In classical topology, a space is called Baire if the intersection of every sequence of dense open sets is dense. Baire spaces play an important role in different areas of mathematics such as analysis and mathematical logic. The concept of Baire spaces has also appeared in fuzzy set theory as well as soft set theory, see [22] and [2]. Fuzzy sets were introduced by Zadeh [23] and soft sets were initially introduced by Molodtsov [13]. Both of these sets were developed to solve the problem of modeling vagueness in real-life problems. Fuzzy sets have been applied in medical diagnosis [1] while fuzzy soft sets have been used to classify wood materials to prevent fire-related injuries and deaths [9]. In bispaces (spaces endowed with two topologies), an almost (i, j)-Baire bispace refers to a bispace (X, τ1, τ2) in which the intersection of any sequence of τi-dense τj-open subsets is τi-open. A study of almost (i, j)-Baire bispaces is documented in [8]. These bispaces also appear in a number of articles such as [7] and [6]. In locale theory, a Baire locale was introduced by Isbell [10] as one in which every non-void open sublocale is of second category. To our knowledge, (i, j)-Baireness has not yet appeared in the category of DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5362 Email address: sibahlezwide@gmail.com (M. Nxumalo) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 2 of 21 bilocales. In this paper, we introduce and study (i, j)-Baire bilocales. Our definition is rather an extension of almost (i, j)-Baire bispaces instead of Baire locales, with the prefix “almost” being dropped. Since the definition of almost (i, j)-Baire bispace is purely in terms of open subsets, we extend it to bilocales almost verbatim. We aim to extend some known bispaces results and also give some natural properties of (i, j)-Baire bilocales. Some of the natural results include (i, j)-Baireness of both the ideal bilocale and the subbilocale induced by the smallest dense sublocale. Extending results from spaces or bispaces to bilocales/biframes is not outrageous. For instance, Schauerte in [21] extended the notion of a normal space to a normal biframe. This paper contributes to the theory of bilocales. This paper is organized as follows. Section two consists of the necessary background. In section three we introduce and characterize (i, j)-Baire bilocales. We also show that the class of (i, j)-Baire bilocales includes the following classes: (i) compact i-prefit bilocales, (ii) bilocales (L,L1, L2) where there is an i-prefit compactification h : (M,M1,M2) → (L,L1, L2) with which h∗[L] is i-Gδ-dense inM , and (iii) i-pseudocomplete bilocales. In the class of Noetherian bilocales, a bilocale is (i, j)-Baire if and only if the induced ideal bilocale is (i, j)-Baire. In section four, we investigate relative versions of (i, j)-Baireness. We show that a bilocale is (i, j)-Baire only if the subbilocale induced by the smallest dense sublocale is (i, j)-Baire. We also introduce and characterize relatively (i, j)-Baire subbilocales. It turns out that in a class of dense subbilocales, (i, j)-Baire coincides with relatively (i, j)- Baire. In section five, we define and characterize (τi, τj)-Baire topobilocales. 2. Preliminaries The book [17] is our main reference for notions of locales and sublocales. See [4, 15, 18] for the theory of bilocales. 2.1. Locales A locale L is a complete lattice in which a ∧ ∨ B = ∨ {a ∧ b : b ∈ B} for all a ∈ L, B ⊆ L. 1L and 0L, with subscripts dropped if there is no possibility of confusion, respectively denote the top element and the bottom element of a locale L. By a point of a locale L we mean an element a of L such that a ̸= 1 and b∧ c ≤ a implies b ≤ a or c ≤ a for all b, c ∈ L. We denote by a∗ the pseudocomplement of an element a ∈ L. An element a ∈ L is said to be dense and complemented in case a∗ = 0 and a ∨ a∗ = 1, respectively. An element x ∈ L is compact if x ≤ ∨ A for A ⊆ L implies x ≤ ∨ F for some finite F ⊆ A. By a compact locale L we mean a locale in which the top element is compact. A regular locale is a locale L in which a = ∨ {x ∈ L : x ≺ a} for every a ∈ L, where x ≺ a means that x∗ ∨ a = 1. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 3 of 21 By a subframe of a locale L, we mean a subset which is closed under joins and finite meets. We denote by OX the locale of open subsets of a topological space X. A localic map is an infima-preserving function f : L → M between locales such that the corresponding left adjoint f∗, called the frame homomorphism, preserves binary meets. A frame homomorphism h : M → L is dense if h(x) = 0 implies x = 0 for all x ∈ M . A sublocale of a locale L is a subset S closed under all meets and x ↠ s ∈ S for every x ∈ L and s ∈ S, where ↠ is a Heyting operation on L satisfying that a ≤ b ↠ c if and only if a ∧ b ≤ c for all a, b, c ∈ L. We denote by O the smallest sublocale of a locale L. We use S(L) to represent the coframe of sublocales of a locale L. For each S ∈ S(L), we define L∖ S := ∨ {T ∈ S(L) : T ∩ S = O}. The sublocales c(a) = {x ∈ L : a ≤ x} and o(a) = {a ↠ x : x ∈ L}, of a locale L are respectively the closed and open sublocales induced by an element a of L. They are complements of each other. The smallest closed sublocale of L that contains a sublocale S is called the closure of S and denoted by S or clL(S) with subscript L dropped when the locale is clear from the context. For every Λ, c (∨ α∈Λ xα ) = ∧ α∈Λ c(xα). A sublocale S of a locale L is dense and nowhere dense if S = L and S ∩ BL = O, respectively, where B(L) = {x ↠ 0 : x ∈ L} is the smallest dense sublocale of L. We refer a reader to [14] for a comprehensive study of nowhere dense sublocales. By a Gδ-sublocale of a locale L, we mean a sublocale of the form S = ∧ n∈N o(xn). For each sublocale S ⊆ L there is an onto frame homomorphism νS : L → S defined by νS(a) = ∧ {s ∈ S : a ≤ s}. Open sublocales and closed sublocales of a sublocale S of L are given by oS(νS(a)) = S ∩ o(a) and cS(νS(a)) = S ∩ c(a), respectively, for a ∈ L. Each localic map f : L → M induces the functions f [−] : S(L) → S(M) given by the set-theoretic image of each sublocale of L under f , and f−1[−] : S(M) → S(L) given by f−1[T ] = ∨ {A ∈ S(L) : A ⊆ f−1(T )}. For a localic map f : L → M and x ∈ M , f−1[cM (x)] = cL(h(x)) and f−1[oM (x)] = oL(h(x)). We denote by à the sublocale of OX induced by a subset A of a topological space X. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 4 of 21 2.2. Bilocales A bilocale is a triple (L,L1, L2) where L1, L2 are subframes of a locale L and for all a ∈ L, a = ∨ {a1 ∧ a2 : a1 ∈ L1, a2 ∈ L2 and a1 ∧ a2 ≤ a}. We call L the total part of (L,L1, L2), and L1 and L2 the first and second parts, respectively. We use the notations Li, Lj to denote the first or second parts of (L,L1, L2), always assuming that i, j = 1, 2, i ̸= j. For every bispace (X, τ1, τ2), there is a corresponding bilocale (τ1 ∨ τ2, τ1, τ2). For example, let (X, τ1, τ2) be a bispace where X = {a, b}, τ1 = {∅, X} and τ2 = {∅, X, {a}}. Then (τ1 ∨ τ2 = τ2, τ1, τ2) is a bilocale. The bilocale pseudocomplement of c ∈ Li is given by c• = ∨ {x ∈ Lj : x ∧ c = 0}. For all a ∈ Lj , b ∈ Li, a ∧ b = 0 if and only if a ≤ b•. A bilocale (L,L1, L2) is compact if its total part is compact, and regular provided that x = ∨ {a ∈ Li : a ≺i x} for every x ∈ Li, where a ≺i x means that there is c ∈ Lj such that a∧c = 0 and c∨x = 1. A biframe homomorphism (or biframe map) h : (M,M1,M2) → (L,L1, L2) is a frame homomorphism h : M → L for which h(Mi) ⊆ Li (i = 1, 2). The map h : M → L is called the total part of h : (M,M1,M2) → (L,L1, L2). By a biframe map we mean a function h : (M,M1,M2) → (L,L1, L2) with a dense total part h : M → L. It is onto if h[Mi] = Li for i = 1, 2. A subbilocale of a bilocale (L,L1, L2) is a triple (S, S1, S2) where S is a sublocale of L and Si = νS [Li] for i = 1, 2. We shall say that (S, S1, S2) is a P -subbilocale in case S has property P . Recall that for a bilocale (L,L1, L2) and a sublocale S of L: [15] inti(S) = ∨ {o(a) : a ∈ Li, o(a) ⊆ S} (i = 1, 2). and [18] cli(S) = ∧ {c(a) : a ∈ Li, S ⊆ c(a)} = c (∨ {a ∈ Li : S ⊆ c(a)} ) (i = 1, 2). For each a ∈ Li, c(a •) = clj(o(a)). A sublocale A of a bilocale (L,L1, L2) is i-dense if cli(A) = L. This is equivalent to saying that S is i-dense if and only if for each non-zero x ∈ Li, o(x) ∩ S ̸= O. Every sublocale containing an i-dense sublocale is i-dense. Given a bilocale (L,L1, L2), a sublocale S of L is (i, j)-nowhere dense if intj(cli(S)) = O (i ̸= j ∈ {1, 2}). As a result, a sublocale S of a bilocale (L,L1, L2) is (i, j)-nowhere dense M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 5 of 21 if and only if L ∖ cli(S) is j-dense if and only if S is (i, j)-nowhere dense. Furthermore, an element a ∈ Li is j-dense if and only if c(a) is (i, j)-nowhere dense. When dealing with subbilocales, say (S, S1, S2), we write iS-dense, iS-open and (iS , jS)- nowhere dense instead of i-dense, i-open and (i, j)-nowhere dense. By an i-Gδ-sublocale of a bilocale (L,L1, L2), we mean a sublocale of the form S =∧ n∈N o(xn) where each xn ∈ Li. A sublocale of a bilocale (L,L1, L2) is i-Gδ-dense if it meets every nonvoid i-Gδ-sublocales. For the bilocale (τ1 ∨ τ2, τ1, τ2) induced by a bispace (X, τ1, τ2), we shall write U is τi-open if U ∈ τi and U is τi-dense if U is dense with respect to the topological space (X, τi). 3. (i, j)-Baire Bilocales Recall that a bispace (X, τ1, τ2) is almost (i, j)-Baire [8] if any collection {Un : n ∈ N} of τi-dense τj-open subsets ofX satisfies the condition ⋂ n∈N Un is τi-dense. In this section, we extend the definition of almost (i, j)-Baire bispaces to bilocales where the prefix “almost” shall be dropped. We aim to define (i, j)-Baire bilocales in such a way that a bispace (X, τ1, τ2) is almost (i, j)-Baire if and only if the bilocale (τ1 ∨ τ2, τ1, τ2) is (i, j)-Baire. We shall call an open (resp. closed) sublocale i-open (resp. i-closed) in case the inducing element is an element of Li. Definition 1. A bilocale (L,L1, L2) is said to be (i, j)-Baire if the intersection of countably many i-dense j-open sublocales is i-dense. Example 1. Since, in locales, the intersection of dense sublocales is dense, every sym- metric bilocale (bilocale of the form (L,L,L)) is (i, j)-Baire. For a bispace (X, τ1, τ2) and A ⊆ X, define à = {intτ1∨τ2((X ∖A) ∪G) : G ∈ τ1 ∨ τ2}. It is clear that à is a sublocale of τ1 ∨ τ2. For each x ∈ X, x̃ = X ∖ clτ1∨τ2 {x} is a point of τ1 ∨ τ2. Just like in the case of locales, o(U) = Ũ for every U ∈ τ1 ∨ τ2. Recall from [11] that given a topological property P , a bispace (X, τ1, τ2) is sup-P if (X, τ1 ∨ τ2) has property P . In [16], we proved the following result. Lemma 1. Let (X, τ1, τ2) be a sup-TD-bispace. Then A ⊆ X is τi-dense in (X, τ1, τ2) iff à is i-dense in (τ1 ∨ τ2, τ1, τ2). In [19], the authors show that if X is a topological space and Y ⊆ X, then Ỹ = ∨ {{X ∖ {y}, 1OX} : y ∈ Y }. Proposition 1. Let (X, τ1, τ2) be a sup-TD-bispace in which Gδ-sublocales of (τ1∨τ2, τ1, τ2) are complemented. Then (X, τ1, τ2) is almost (i, j)-Baire iff (τ1 ∨ τ2, τ1, τ2) is (i, j)-Baire. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 6 of 21 Proof. (=⇒): Let {o(Un) : n ∈ N} be a collection of i-dense j-open sublocales. It follows that {Un : n ∈ N} is a collection of τi-dense τj-open subsets of X. It follows that⋂ n∈N Un is τi-dense. Claim: ∧ n∈N o(Un) is i-dense. Proof: Let o(V ) be an i-open sublocale such that o(V ) ∩ ∧ n∈N o(Un) = O. Then∧ n∈N o(V ∩ Un) = O. We must have that ⋂ n∈N(V ∩ Un) = ∅. Otherwise, there is x ∈ V ∩ Un for each n ∈ N. Since each V ∩ Un is τ1 ∨ τ2, Ṽ ∩ Un = o(V ∩ Un) ∋ x̃ for each n ∈ N. Therefore x̃ ∈ ∧ n∈N o(V ∩ Un) which is impossible. Therefore V = ∅ so that o(V ) = O. Thus ∧ n∈N o(Un) is i-dense. (⇐=): Let {Un : n ∈ N} be a collection of τi-dense τj-open subsets of X. Then {o(Un) : n ∈ N} is a collection of i-dense j-open sublocales of τ1 ∨ τ2. It follows that∧ n∈N o(Un) is τi-dense. To show that ⋂ n∈N Un is τi-dense, let V be a nonempty τi-open subset of X such that V ∩ (⋂ n∈N Un ) = ∅. Then⋃ n∈N (X ∖ (V ∩ Un)) = X. Observe that ∨ n∈N c(V ∩ Un) = OX: Let p ∈ X. Then p ∈ X ∖ (V ∩ Un) for some n ∈ N. Therefore {p} ⊆ X ∖ (V ∩ Un) for some n ∈ N so that V ∩ Un ⊆ X ∖ {p}. This implies that X ∖ {p} ∈ c(V ∩ Un). As a result, {X ∖ {p}, 1τ1∨τ2} ⊆ c(V ∩ Un). Therefore τ1 ∨ τ2 = ∨ {{X ∖ {p}, 1τ1∨τ2} : p ∈ X} ⊆ ∨ {c(V ∩ Un) : n ∈ N} ⊆ τ1 ∨ τ2. Since ∧ n∈N o(Un) is i-dense and o(V ) is non-void i-open, o(V ) ∩ ∧ n∈N o(Un) ̸= O, so that O ̸= ∧ n∈N o(V ∩ Un). M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 7 of 21 Because ∧ n∈N o(V ∩ Un) is a complemented Gδ-sublocale, we have that∨ n∈N c(V ∩ Un) ̸= τ1 ∨ τ2, which is impossible. Definition 2. Let (L,L1, L2) be a bilocale. A sublocale S of L is said to be of (i, j)-first category if there are countably many (i, j)-nowhere dense sublocales Sn, n ∈ N, such that S ⊆ ∨ n∈N Sn. It is of (i, j)-second category if it is not of (i, j)-first category. Theorem 1. Let (L,L1, L2) be a bilocale whose j-Gδ-sublocales are complemented in L. The following statements are equivalent: (i) (L,L1, L2) is (i, j)-Baire. (ii) Each non-void i-open sublocale is of (j, i)-second category. (iii) Every sublocale of (j, i)-first category has void i-interior. (iv) The supplement of every sublocale of (j, i)-first category is i-dense. Proof. (i) =⇒ (ii): Let U be a non-void i-open sublocale of L and assume that U ⊆ ∨ n∈N Sn for some collection {Sn : n ∈ N} of (j, i)-nowhere dense sublocales. Then U ⊆ ∨ n∈N Sn where members of the collection {Sn : n ∈ N} are (j, i)-nowhere dense. It follows that members of the collection {L∖ clj(Sn) : n ∈ N} are i-dense j-open sublocales. By hypothesis, ∧ n∈N(L∖ clj(Sn) is i-dense so that U ∩ (∧ n∈N (L∖ clj(Sn)) ) ̸= O. Therefore O ̸= (∨ k∈N Sk ) ∩ (∧ n∈N (L∖ clj(Sn)) ) = ∨ k∈N ( Sk ∩ (∧ n∈N (L∖ Sn) )) ⊆ ∨ k∈N ( Sk ∩ (L∖ Sk) ) = O which is impossible. (ii) =⇒ (iii): Let S be a sublocale of (j, i)-first category with a non-void i-interior. We then get that inti(S) is a non-void i-open sublocale which must be of (j, i)-second category by (ii). This is a contradiction. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 8 of 21 (iii) =⇒ (iv): Let S be a sublocale of L which is of (j, i)-first category and choose x ∈ Li with o(x) ∩ (L ∖ S) = O. Then o(x) ⊆ S. Since S satisfies the conditions hypothesized in (iii), inti(S) ̸= O, so that the i-open sublocale o(x) is void. (iv) =⇒ (i): Let {o(xn) : n ∈ N} be a collection of i-dense j-open sublocales and assume that there is an i-open sublocale o(y) with o(y) ∩ (∧ n∈N o(xn) ) = O Then o(y) ⊆ ∨ n∈N c(xn), making o(y) a sublocale of (j, i)-first category. By (iv), L∖o(y) = c(y) is i-dense, i.e., cli(c(y)) = c(y) = 1. Therefore y = 0 so that o(y) = O. Hence(∧ n∈N o(xn) ) is i-dense. We shall say that a collection C of sublocales of L has the Finite Intersection Property (FIP) if the intersection of every finite subcollection of C has a non-void intersection. Proposition 2. If a bilocale (L,L1, L2) is compact, then every collection of closed sublo- cales with the FIP has a non-void intersection. Proof. Let {c(xα) : α ∈ Λ} be a collection with the FIP and assume that ∧ α∈Λ c(xα) = O. Then L = L∖ ∧ α∈Λ c(xα) = ∨ α∈Λ o(xα), making o (∨ α∈Λ xα ) = L. Therefore ∨ α∈Λ xα = 1. Since (L,L1, L2) is compact, there is a finite set F ⊆ Λ such that ∨ α∈F xα = 1. We get that O = c (∨ α∈F xα ) = ∧ α∈F c(xα), which contradicts that {c(xα) : α ∈ Λ} has the FIP. Remark 1. The converse of the preceding result holds. We are however interested in the forward direction, that is why we only proved it. Recall from [20] that a locale L is prefit if for each nonzero x ∈ L there is a nonzero y ∈ L such that y⋆ ∨ x = 1. A bispace (X, τ1, τ2) is almost regular if for each nonempty U ∈ τi, there is nonempty V ∈ τi such that clτj (V ) ⊆ U . Since prefitness is a localic version of almost regularity in spaces (spaces in which every nonempty open set contains some closure of a nonempty open subset), we define a prefit bilocale (L,L1, L2) using the notion of almost regular bispace as one in which for each x ∈ Li, i = 1, 2, there is y ∈ Li such that y• ∨ x = 1. Related to prefit bilocales, we give the following definition. Definition 3. Call a bilocale (L,L1, L2) i-prefit in case for every nonzero x ∈ L, there is a nonzero y ∈ Li such that y• ∨ x = 1. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 9 of 21 We consider some examples. Example 2. (i) The bilocale of reals is an example of a prefit bilocale which is not i-prefit. (ii) For any almost regular bispace (X, τ1, τ2) with τ1 ⊆ τ2, the bilocale (τ1 ∨ τ2, τ1, τ2) is 2-prefit. In particular, if L is prefit, then (L,L,L) is i-prefit (i = 1, 2). (iii) By [21], a bilocale (L,L1, L2) is Boolean if for each x ∈ Li, i = 1, 2, there is c ∈ Lj (i ̸= j) such that x ∧ c = 0 and x ∨ c = 1. Boolean and i-prefit are incomparable: Consider the set X = {a, b, c, d} endowed with topologies τ1 = {∅, X, {a}, {b}, {a, b}} and τ2 = {∅, X, {b, c, d}, {a, c, d}, {c, d}}. It is clear that (τ1 ∨ τ2, τ1, τ2) is Boolean. This bilocale is not i-prefit (i = 1, 2) since for the set {a} ∈ τ1 ∨ τ2, there is no nonempty U ∈ τi satisfying that U• ∨ {a} = X. For any non-Boolean prefit locale L, (L,L,L) is an example of a non-Boolean i-prefit (i = 1, 2) bilocale. In the following result, we show that the class of (i, j)-Baire bilocales contains compact i-prefit bilocales. Proposition 3. Every compact i-prefit bilocale is (i, j)-Baire. Proof. Let (L,L1, L2) be a compact i-prefit bilocale and choose a collection {o(xn) : n ∈ N, xn ∈ Li} of i-dense j-open sublocales and a non-void j-open sublocale o(y). Then o(y) ∩ o(xn) ̸= O for each n ∈ N. This makes y∧xn ̸= 0. Since (L,L1, L2) is i-prefit, there is nozero b1 ∈ Li such that b•1 ∨ (y ∧ x1) = 1. Because o(x2) is i-dense, we have that o(x2) ∩ o(b1) ̸= O so that x2 ∧ b1 is a nonzero element of L. By i-prefitness again, there is nonzero b2 ∈ Li such that b•2 ∨ (x2 ∧ b1) = 1. Continuing like this for n = 3, 4, .., we find bn ∈ Li such that b•n ∨ (xn ∧ bn−1) = 1. Therefore c(b•n) ⊆ o(xn) ∩ o(bn−1). Since each bn ∈ Li, we have that c(b•n) = clj(o(bn)). Therefore ... ⊆ c(b•3) = clj(o(b3)) ⊆ c(b•2) = clj(o(b2)) ⊆ c(b•1) = clj(o(b1)) ⊆ o(y) ∩ o(x1). We now have the decreasing sequence c(b•1), c(b • 2), c(b • 3), ... of closed sublocales, so that the collection {c(b•n) : n ∈ N} has the FIP. By compactness of (L,L1, L2), ∧ n∈N c(b•n) ̸= O. Because ∧ n∈N c(b•n) ⊆ o(y) ∩ ∧ n∈N o(xn), we then have that o(y) ∩ ∧ n∈N o(xn) ̸= O, making ∧ n∈N o(xn) an i-dense sublocale. The converse of Proposition 3 is not always true, as shown below. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 10 of 21 Example 3. For the bispace (N, τD, τcf ), (τD ∨ τcf = τD, τD, τcf ) is (τD, τcf )-Baire but not compact. Its (τD, τcf )-Baireness follows since N is the only τD-dense member of τcf . Recall that for any onto frame homomorphism h : M → L, h∗ : L → h∗[L] is a frame isomorphism. If h is further dense, then h∗(0) = 0. Lemma 2. Let h : (M,M1,M2) → (L,L1, L2) be a dense and onto biframe map. If x ∈ Li is j-dense, then h∗(x) ∧ a ̸= 0 whenever a ∈ Mi is nonzero. Proof. Let a ∈ Mi be nonzero such that a ∧ h∗(x) = 0. Then 0 = h(a) ∧ h(h∗(x)) = h(a) ∧ x. Since h[Mi] ⊆ Li, h(a) ∈ Li so that h(a) = 0. Therefore a ≤ h∗(h(a)) = h∗(0) = 0. Proposition 4. Let (L,L1, L2) be a bilocale. If there is a dense onto biframe map h : (M,M1,M2) → (L,L1, L2) from an (i, j)-Baire bilocale (M,M1,M2) with which h∗[L] is i-Gδ-dense in M , then (L,L1, L2) is (i, j)-Baire. Proof. Suppose that the hypothesized statement is true and let {o(xn) : n ∈ N} be a collection of i-dense j-open sublocales of L. Now, if o(y) ∩ (∧ n∈N o(xn) ) = O for some i-open sublocale o(y) of L, then O = h∗[o(y)] ∩ h∗ [∧ n∈N o(xn) ] = h∗[o(y)] ∩ (∧ n∈N h∗[o(xn)] ) where the first equality follows since h∗[O] = O and h∗ is injective, and the second equality follows since the total part h∗ : L → M is a right adjoint. By virtue of h∗ : L → h∗[L] being a frame isomorphism and hence open, we get that oh∗[L](h∗(y)) ∩ (∧ n∈N oh∗[L](h∗(xn)) ) = O. For each n ∈ N, h∗(xn) = h∗(h(an)) for some an ∈ Mi. Therefore O = oh∗[L](h∗(y)) ∩ (∧ n∈N oh∗[L](h∗(h(an))) ) = h∗[L] ∩ o(h∗(y)) ∩ (∧ n∈N (h∗[L] ∩ o(h∗(h(an)))) ) = h∗[L] ∩ o(h∗(y)) ∩ (∧ n∈N o(h∗(h(an))) ) ⊇ h∗[L] ∩ o(h∗(y)) ∩ (∧ n∈N o(an) ) . M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 11 of 21 Since i-Gδ-dense sublocales are dense, h∗[L] is a dense sublocale of M . Each of the an’s is i-dense: Pick b ∈ Mi such that b ∧ an = 0. Then h(b) ∧ h(an) = 0 so that h∗(h(b)) ∧ h∗(h(an)) = h∗(0) = 0, where the latter equality follows since h∗ : L → M is a dense localic map. Therefore b ∧ h∗(xn) = 0. By Lemma 2, b = 0 and hence each an is i-dense. Therefore the collection {o(an) : n ∈ N} consists of i-dense j-open sublocales of M . We then get that o(h∗(y))∩ (∧ n∈N (o(an)) ) is an i-Gδ-sublocale. Because h∗[L] is i-Gδ-dense, so o(h∗(y)) ∩ (∧ n∈N o(an) ) = O. Since (M,M1,M2) is (i, j)-Baire, it follows that ∧ n∈N o(an) is i-dense, so that o(h∗(y)) = O. Therefore h∗(y) = 0, so that 0 = h(h∗(y)) = y. This means that o(y) = O. Thus ∧ n∈N o(xn) is i-dense, and hence (L,L1, L2) is (i, j)- Baire. A compactification of a bilocale (L,L1, L2) is a dense and onto biframe map h : (M,M1,M2) → (L,L1, L2) from a compact regular bilocale (M,M1,M2). So, for a bilo- calic property P , we shall say that (L,L1, L2) has a P -compactification in case (M,M1,M2) has property P . Corollary 1. Let (L,L1, L2) be a bilocale. If there is a i-prefit compactification h : (M,M1,M2) → (L,L1, L2) with which h∗[L] is i-Gδ-dense in M , then (L,L1, L2) is (i, j)- Baire. Definition 4. Let (L,L1, L2) be a bilocale. An i-π-base for (L,L1, L2) is a collection C of non-void i-open sublocales such that each non-void i-open sublocale of L contains a member of C. A bilocale is said to be i-pseudocomplete if it is i-prefit and it has a sequence (Cn)n∈N of i-π-bases such that whenever o(xn) ∈ Cn and clj(o(xn+1)) ⊆ o(xn) for each n, then ∧ n∈N o(xn) ̸= O. Proposition 5. Every i-pseudocomplete bilocale is (i, j)-Baire. Proof. Let (L,L1, L2) be a pseudocomplete bilocale and pick a collection {o(xn) : n ∈ N} of i-dense j-open sublocales. Since (L,L1, L2) is pseudocomplete, there is a sequence (Cn)n∈N of i-π-bases with the corresponding pseudocompleteness property. For each non- void i-open sublocale o(y), we have that each o(y) ∩ o(xn) is a non-void open sublocale, so that y ∧ xn ̸= 0. Since (L,L1, L2) is i-prefit, for n = 1, there is nonzero a1 ∈ Li such that a•1 ∨ (y ∧ x1) = 1. This makes O ̸= o(a1) ⊆ clj(o(a1)) ⊆ o(y) ∩ o(x1). M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 12 of 21 For the i-π-base C1, there is o(c1) ∈ C1 with o(c1) ⊆ clj(o(c1)) ⊆ o(a1) ⊆ o(y) ∩ o(x1). Using the fact that o(x2) is i-dense and o(c1) is non-void i-open, we get that o(c1)∩ o(x2) is a non-void open sublocale. Since (L,L1, L2) is i-prefit, there is nonzero a2 ∈ Li such that O ̸= o(a2) ⊆ clj(o(a2)) ⊆ o(c1) ∩ o(x2). An application of pseudocompleteness to the i-π-base C2 yields an existence of o(c2) ∈ C2 such that o(c2) ⊆ clj(o(c2)) ⊆ o(a2) ⊆ o(c1) ∩ o(x2). Since o(x3) is i-dense and o(c2) is a non-void i-open sublocale, it follows that o(c2)∩o(x3) ̸= O. Applying that (L,L1, L2) is i-prefit again implies that there is a nonzero a3 ∈ Li such that O ̸= o(a3) ⊆ cli(o(a3)) ⊆ o(c2) ∩ o(x3). Therefore, for the i-π-base C3, there is o(c3) ∈ C3 such that o(c3) ⊆ clj(o(c3)) ⊆ o(a3) ⊆ o(c2) ∩ o(x3). Continuing like this for n = 4, 5, ...., we get that for each i-π-base Cn, there is o(cn) ∈ Cn such that o(cn) ⊆ clj(o(cn)) ⊆ o(cn−1) ∩ o(xn). Since (L,L1, L2) is i-pseudocomplete, ∧ n∈N o(cn) ̸= O. Because o(cn) ⊆ o(y) ∩ o(xn) for each n ∈ N, we have O ̸= ∧ n∈N o(cn) ⊆ o(y) ∩ ∧ n∈N o(xn), making ∧ n∈N o(xn) an i-dense sublocale. Thus (L,L1, L2) is (i, j)-Baire. Recall from [5] that the triple (JL, (JL)1, (JL)2), where JL is the locale of all ideals of L and (JL)i (i = 1, 2) is the subframe of JL consisting of all ideals J ⊆ L generated by J ∩ Li, is a bilocale called the ideal bilocale. Call a bilocale (L,L1, L2) Noetherian in case its total part L is Noetherian, i.e., all of its elements are compact. In a Noetherian locale, all ideals are principal [3]. This suggests that in a Noetherian locale, the locale JL of ideals of a locale L is isomorphic to L. For use below, we recall from [15, Proposition 6.9.] that in a bilocale (L,L1, L2), if x ∈ Li is j-dense, then ↓x ∈ JLi is JLj-dense. Furthermore, for a Noetherian bilocale (L,L1, L2), ∨ J ∈ Li is j-dense whenever J ∈ JLi is JLj-dense. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 13 of 21 Proposition 6. Let (L,L1, L2) be a bilocale. Then (L,L1, L2) is (i, j)-Baire only if (JL, (JL)1, (JL)2) is (i, j)-Baire. Moreover, if (L,L1, L2) is Noetherian, then (L,L1, L2) is (i, j)-Baire iff (JL, (JL)1, (JL)2) is (i, j)-Baire. Proof. Choose a collection {o(xn) : n ∈ N} of i-dense j-open sublocales. Then {oJL(↓xn) : n ∈ N} is a collection of JLi-dense JLj-open sublocales. By hypothesis, ∧ n∈N oJL(↓xn) is JLi-dense. To show that ∧ n∈N o(xn) is i-dense, let o(y) be an i-open sublocale such that o(y) ∩ (∧ n∈N o(xn) ) = O. Claim: oJL(↓y) ∩ (∧ n∈N oJL( ∨ ↓xn) ) = O. Proof: Otherwise, ∧ n∈N o(↓y ∩ ↓xn) ̸= O which implies that O ̸= ↓y ∩ ↓xn = ↓(y ∧ xn) for each n ∈ N. Therefore y ∧ xn ̸= 0 for each n ∈ N so that O ̸= ∧ n∈N o(y ∧ xn) = o(y) ∩ (∧ n∈N o(xn) ) which is a contradiction. Thus oJL(↓y) = O implying that ↓y = O. Therefore o(y) = O and hence ∧ n∈N o(xn) is i-dense. The particular case follows since L is isomorphic to JL. 4. Concerning relative versions of (i, j)-Baire bilocales In this section, we consider (i, j)-Baireness of subbilocales. We recall the following lemma from [16]. Lemma 3. Let (S, S1, S2) be a dense subbilocale of a bilocale (L,L1, L2). An element y of Li is j-dense iff νS(y) is jS-dense. Corollary 2. Let (S, S1, S2) be a dense subbilocale of a bilocale (L,L1, L2). An element y of Li is j-dense iff oS(νS(y)) = S ∩ o(y) is jS-dense iS-open. We also have the following result. Lemma 4. Let (L,L1, L2) be a bilocale with (S, S1, S2) as its dense subbilocale. A sublocale A of S is iS-dense iff it is i-dense. Proof. (=⇒): Choose a non-void i-open sublocale o(x) of L. Then O ̸= S ∩ o(x) = o(νS(x)) where νS(x) ∈ Si. This makes oS(νS(x)) an non-void iS-open sublocale of S. Since A is iS-open, O ̸= A ∩ oS(νS(x)) = A ∩ o(x). M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 14 of 21 Thus A is i-dense. (⇐=): Let oS(x) be a non-void iS-open sublocale of S. Then x = νS(y) for some y ∈ Li. It follows from Lemma 3 that y is i-dense. Therefore O ̸= A ∩ o(y) = A ∩ oS(x). Thus A is iS-dense. Proposition 7. A bilocale (L,L1, L2) is (i, j)-Baire only if it contains some dense (i, j)- Baire subbilocale. Proof. Let (S, S1, S2) be a dense and (i, j)-Baire subbilocale of (L,L1, L2) and pick a collection {o(xn) : n ∈ N} of i-dense j-open sublocales. Since the subbilocale (S, S1, S2) is dense, it follows from Corollary 2 that {S ∩ o(xn) : n ∈ N} is a collection of iS-dense jS-open sublocales. By hypothesis, ∧ n∈N(S ∩ o(xn)) is iS-dense, so that it is i-dense by Lemma 4. Since ∧ n∈N (S ∩ o(xn)) ⊆ ∧ n∈N o(xn), it follows that ∧ n∈N o(xn) is i-dense. Corollary 3. A bilocale (L,L1, L2) is (i, j)-Baire only if (BL, νB[L1], νB[L2]) is (i, j)- Baire as a bilocale. Call a bilocale (L,L1, L2) (i, j)-submaximal if every i-dense sublocale of L is j-open Proposition 8. Let (L,L1, L2) be an (i, j)-submaximal bilocale. Then (L,L1, L2) is (i, j)- Baire iff (BL, νB[L1], νB[L2]) is (i, j)-Baire as a bilocale. Proof. We only prove the forward implication: Let {oBL(xn) : n ∈ N} be a collection of iBL-dense jBL-open sublocales. It follows that {o(xn) : n ∈ N} is a collection of i-dense j-open sublocales. Since (L,L1, L2) is (i, j)-Baire,∧ n∈N o(xn) is i-dense. We must have that ∧ n∈N oBL(xn) is iBL-dense, otherwise there is a non-void νBL[Li]-open sublocale oBL(y) such that oBL(y) ∩ (∧ n∈N oBL(xn) ) = O. Therefore oBL(y) ∩ (∧ n∈N o(xn) ) = O. Since every dense sublocale is i-dense and (L,L1, L2) is (i, j)-submaximal, we have that BL is j-open so that oBL(y) = BL ∩ o(y) is a j-open sublocale. Therefore oBL(y) = 0 which is impossible. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 15 of 21 Proposition 9. Every i-open subbilocale of an (i, j)-Baire bilocale is (i, j)-Baire. Proof. Let (S, S1, S2) be an i-open subbilocale of an (i, j)-Baire bilocale (L,L1, L2). Choose a collection {oS(xn) : n ∈ N} of iS-dense jS-open sublocales. We show that∧ n∈N oS(xn) is iS-dense. Pick an iS-open sublocale oS(y) such that(∧ n∈N oS(xn) ) ∩ oS(y) = O. Since oS(y) ⊆ S, oS(y) ∩ (L∖ S) = O. Therefore oS(y) ∩ ((∧ n∈N oS(xn) ) ∨ (L∖ S) ) = ((∧ n∈N oS(xn) ) ∩ oS(y) ) ∨ ( oS(y) ∩ (L∖ S) ) = (∧ n∈N oS(xn) ) ∩ oS(y) = O. Because oS(xn) ∨ (L∖ S) is i-dense, it follows that∧ n∈N ( oS(xn) ∨ (L∖ S) ) = (L∖ S) ∨ ∧ n∈N oS(xn) is i-dense. Therefore oS(y) = O. Thus ∧ n∈N oS(xn) is iS-dense. Definition 5. Let (L,L1, L2) be a bilocale. A subbilocale (S, S1, S2) of (L,L1, L2) is relatively (i, j)-Baire if for every collection {o(xn) : n ∈ N} of i-dense j-open sublocales, S ∩ (∧ n∈N o(xn) ) is iS-dense. Proposition 10. In a class of dense subbilocales, (i, j)-Baire coincides with relatively (i, j)-Baire. Proof. Let (S, S1, S2) be an (i, j)-Baire subbilocale of a bilocale (L,L1, L2) and choose a collection {o(xn) : n ∈ N} of i-dense j-open sublocales of L. If oS(y) ∩ S ∩ (∧ n∈N o(xn) ) = O, then O = oS(y) ∩ (∧ n∈N (S ∩ o(xn)) ) = oS(y) ∩ (∧ n∈N oS(νS(xn)) ) where each oS(νS(xn)) is iS-dense and jS-open. Since (S, S1, S2) is (i, j)-Baire as a bilocale,∧ n∈N oS(νS(xn)) is iS-dense so that oS(y) = O. Thus S ∩ (∧ n∈N o(xn) ) is iS-dense. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 16 of 21 On the other hand, let (S, S1, S2) be a relatively (i, j)-Baire subbilocale and pick a collection {oS(xn) : n ∈ N} of iS-dense jS-open sublocales of S. For each xn, there is an ∈ Lj such that xn = νS(an). Now, members of the collection {oS(an) : n ∈ N} are i-dense j-open in (L,L1, L2). Since (S, S1, S2) is relatively (i, j)-Baire, S ∩ (∧ n∈N o(an) ) = ∧ n∈N oS(xn) is iS-dense. Thus (S, S1, S2) is (i, j)-Baire. Here is an example of what is illustrated in Proposition 10. Example 4. Given a bilocale (L,L1, L2), the subbilocale (BL, νB[L1], νB[L2]) of (L,L1, L2) is (i, j)-Baire if and only if it is relatively (i, j)-Baire. We close this section with a characterization of relatively (i, j)-Baire subbilocales. Proposition 11. Let (S, S1, S2) be a dense and complemented subbilocale of a bilocale (L,L1, L2) whose j-Gδ-sublocales are complemented. The following statements are equiv- alent: (i) (S, S1, S2) is relatively (i, j)-Baire. (ii) For every non-void i-open sublocale U of L, S ∩ U is of (j, i)-second category in (S, S1, S2). (iii) For every sublocale U of (j, i)-first category in (L,L1, L2), intiS (S ∩ U) = O. (iv) If V is a sublocale of (j, i)-first category in (L,L1, L2), then S ∩ (L∖V ) is iS-dense. Proof. (i) =⇒ (ii): Let o(x) be non-void i-open and assume that S ∩ o(x) ⊆ S∨ n∈N cS(xn) for some collection {cS(xn) : n ∈ N} of (jS , iS)-nowhere dense sublocales of S. Then S ∩ o(x) ⊆ ∨ n∈N c(xn) where each c(xn) is (j, i)-nowhere dense because (S, S1, S2) is dense. It is clear that the collection {o(xn) : n ∈ N} consists of i-dense j-open sublocales. It follows from (i) that S∩ (∧ n∈N o(xn) ) is iS-dense. Since S∩o(x) ̸= O because of density of (S, S1, S2), we have that S ∩ o(x) ∩ S ∩ (∧ n∈N o(xn) ) ̸= O. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 17 of 21 Therefore (∨ k∈N c(xk) ) ∩ (∧ n∈N o(xn) ) ̸= O. Since ∧ n∈N o(xn) is a j-Gδ-sublocale of L, it follows that it is complemented. Therefore O ̸= ∨ k∈N ( c(xk) ∩ (∧ n∈N o(xn) )) ⊆ ∨ k∈N (c(xk) ∩ o(xk)) = ∨ n∈N (O) = O which is impossible. Thus S ∩ o(x) is (j, i)-second category. (ii) =⇒ (i): Let {o(xn) : n ∈ N} be a collection of i-dense j-open sublocales and assume that there is non-void iS-open sublocale oS(y) of S such that oS(y) ∩ S ∩ (∧ n∈N o(xn) ) = O. Then o(y) is non-void i-open and oS(y) ∩ (∧ n∈N o(xn) ) = O which implies oS(y) ⊆ S ∩ (∨ n∈N o(xn) ) = ∨ n∈N cS(νS(xn)) where the latter equality holds since S is complemented. Since each cS(νS(xn)) is (jS , iS)- nowhere dense, S ∩ o(y) = oS(y) is of (j, i)-first category in (S, S1, S2) which is a contra- diction. (ii) =⇒ (iii): Let U be a sublocale of L which is of (j, i)-first category in (L,L1, L2) and assume that intiS (S ∩ U) ̸= O. Then intiS (S ∩ U) = o(x) ∩ S for some x ∈ Li. Such o(x) is a non-void i-open sublocale of L, so intiS (S ∩ U) = S ∩ o(x) must be of (j, i)-second category in (S, S1, S2) by (ii). But U ⊆ ∨ n∈N c(xn) for some collection {c(xn) : n ∈ N} of (j, i)-nowhere dense sublocales of L, so intiS (S ∩ U) = o(x) ∩ S ⊆ U ∩ S ⊆ S ∩ ∨ n∈N c(xn) = ∨ n∈N cS(νS(xn)) where each cS(νS(xn)) is (jS , iS)-nowhere dense in (S, S1, S2). This makes o(x) ∩ S a sublocale of (j, i)-first category in (S, S1, S2) which is impossible. M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 18 of 21 (iii) =⇒ (iv): Let V be a sublocale of L which is of (j, i)-first category in (L,L1, L2) and choose an iS-open sublocale oS(x) such that oS(x) ∩ S ∩ (L∖ V ) = O. Then oS(x) ⊆ S ∩ V. By (iii), intiS(S ∩ V ) = O, making oS(x) = O. (iv) =⇒ (ii): Let o(x) be a non-void i-open sublocale of L and assume that S ∩ o(x) is of (j, i)-first category. By (iv), S ∩ (L∖ o(x)) = S ∩ c(x) = cS(νS(x)) is iS-dense which implies that νS(x) = 0. Therefore o(x) = O which is a contradiction. 5. Baireness of topobilocales The aim of this section is to introduce and characterize Baireness in the category of topobilocales. A topobilocale [12] is a triple (L, τ1, τ2) where L is a locale, L1 and L2 are subframes of L all of whose elements are complemented in L. Each member of τi (i = 1, 2) is called τi-open. For each a ∈ L, the τi-closure (i = 1, 2) of a in L is defined by cl(L,τi)(a) = ∧ {b ∈ τ ′i : a ≤ b} and the τi-interior of a is defined by int(L,τi)(a) = ∨ {b ∈ τi : b ≤ a}. We have the following result. See [24] for the proofs of some of the statements. For the rest of the statements, the proofs resemble that of [16, Proposition 5.1.3.]. Proposition 12. Let (L, τi, τj) be a topobilocale. Then (i) cl(L,τi)(0) = int(L,τi)(0) = 0. (ii) cl(L,τi)(1) = int(L,τi)(1) = 1. (iii) a ≤ cl(L,τi)(a). (iv) If a ≤ b, then cl(L,τi)(a) ≤ cl(L,τi)(b). (v) int(L,τi)(a) ≤ a. (vi) If a ≤ b, then int(L,τi)(a) ≤ int(L,τi)(b). M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 19 of 21 (vii) For each a ∈ L, (cl(L,τi)(a)) ′ = int(L,τi)(a ′). (viii) For each a ∈ L, (int(L,τi)(a)) ′ = cl(L,τi)(a ′). Call an element a ∈ L τi-dense if cl(L,τi)(a) = 1. Clearly, a ∈ L is τi-dense if and only if a ∧ x ̸= 0 for all nonzero x ∈ τi, see [15, Proposition 2.8.] for the proof. Definition 6. Call a topobilocale (L, τ1, τ2) (τi, τj)-Baire if any sequence (xn)n∈N of τi- dense elements of τj satisfies the condition ∧ n∈N xn is τi-dense. Call an element a ∈ L (τi, τj)-nowhere dense if int(L,τj)(cl(L,τi)(a)) = 0. An element a ∈ L is of (τi, τj)-first category if a ≤ ∨ n∈N xn for some collection {xn : n ∈ N} of (τi, τj)-nowhere dense elements of L. Otherwise it is of (τi, τj)-second category. It is clear that if a is of (τi, τj)-first category and b ≤ a, then b is of (τi, τj)-first category. For use below, we give the following result with a proof similar to that of [16, Propo- sition 2.1.4.] Proposition 13. Let (L, τ1, τ2) be a topobilocale. Then a ∈ L is (τi, τj)-nowhere dense iff (cl(L,τi)(a)) ′ is τj-dense. Proposition 14. Let (L, τ1, τ2) be a topobilocale. The following statements are equivalent. (i) (L, τ1, τ2) is (τi, τj)-Baire. (ii) Each nonzero τi element is of (τj , τi)-second category. (iii) Every element of (τj , τi)-first category has a zero τi-interior. (iv) The complement an element of (τj , τi)-first category is τi-dense. Proof. (i) =⇒ (ii): Assume that there is a nonzero element a ∈ τi which is of (τj , τi)- first category. Then a ≤ ∨ n∈N xn for some collection {xn : n ∈ N} of (τj , τi)-nowhere dense elements. It is clear members of the collection {(cl(L,τj)(xn))′ : n ∈ N} are τi-dense. By (i), ∧ n∈N(cl(L,τj)(xn)) ′ is τi-dense. It follows that a ∧ (∧ n∈N (cl(L,τj)(xn)) ′ ) ̸= 0. Therefore 0 ̸= (∨ k∈N xk ) ∧ (∧ n∈N (cl(L,τj)(xn)) ′ ) = ∨ k∈N ( xk ∧ (∧ n∈N (cl(L,τj)(xn)) ′ )) since L is a locale ≤ ∨ k∈N ( xk ∧ (cl(L,τj)(xk)) ′ ) M. Nxumalo / Eur. J. Pure Appl. Math, 18 (1) (2025), 5362 20 of 21 ≤ ∨ k∈N ( cl(L,τi)(xk) ∧ cl(L,τj)(xk) ) = 0 which is a contradiction. (ii) ⇒ (iii): Let a ∈ L be of (τj , τi)-first category and assume that int(L,τi)(a) ̸= 0. We now have int(L,τi)(a) as a nonzero τi element. It follows from (ii) that int(L,τi)(a) is of (τj , τi)-second category. This is not possible. (iii) ⇒ (iv): Let a ∈ L be of (τj , τi)-first category and suppose that a′ is not τi-dense. Then cl(L,τi)(a ′) ̸= 1. Because cl(L,τi)(a ′) = (int(L,τi)(a)) ′, we have that (int(L,τi)(a)) ′ ̸= 1. Since a is of (τj , τi)-first category, it follows from (iii) that int(L,τi)(a) = 0 so that (int(L,τi)(a)) ′ = 1, which is a contradiction. (iv) ⇒ (i): Let (xn)n∈N be a sequence of τi-dense elements of τj and assume that there is y ∈ τi such that y ∧ (∧ n∈N xn ) = 0. Then y ≤ (∧ n∈N xn )′ = ∨ n∈N x′n since each xn is complemented. This makes y to be of (τj , τi)-first category. By (iv), y′ is τi-dense so that y = 0. Thus ∧ n∈N xn is τi-dense. Hence (L, τ1, τ2) is (τi, τj)-Baire. Acknowledgements The author is indebted to the referee for helpful comments. References [1] K. P. Adlassnig. Fuzzy set theory in medical diagnosis. IEEE Transactions on Sys- tems, Man, and Cybernetics, 16(2):260–265, 1986. [2] Z.A. Ameen and A.B. Khalaf. The invariance of soft baire spaces under soft weak functions. 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