@ Appl. Gen. Topol. 17, no. 1(2016), 7-16 doi:10.4995/agt.2016.3919 c© AGT, UPV, 2016 A uniform approach to normality for topological spaces Ankit Gupta a and Ratna Dev Sarma b a Department of Mathematics, University of Delhi, Delhi 110007, India. (ankitsince1988@yahoo.co.in) b Department of Mathematics, Rajdhani College, University of Delhi, Delhi 110015, India. (ratna sarma@yahoo.com) Abstract (λ,µ)-regularity and (λ,µ)-normality are defined for generalized topo- logical spaces. Several variants of normality existing in the litera- ture turn out to be particular cases of (λ,µ)-normality. Uryshon’s lemma and Tietze extension theorem are discussed in the light of (λ,µ)- normality. 2010 MSC: 54A05; 54D10. Keywords: generalized topology; normality; regularity. 1. Introduction A large amount of research in topology is devoted to the study of classes of subsets of topological spaces, which posses properties similar to those of open sets. In the literature, several such classes are available which include amongst others semi-open sets [10], α-open sets [12], β-open sets [1], pre-open sets [11], etc. Some other such classes are A-sets [15], B-sets [16], C-sets [7], etc. Since these classes have some features common in them, it is quite natural to enquire if these classes can be obtained by using one common definition? Á. Császàr has successfully provided an answer in this regard. The main tool he has used is, the class of mappings γ : P(X) → P(X) from the power set X into X itself possessing the property of monotonicity (that is, for A ⊆ B implies γ(A) ⊆ γ(B)). In a topological space (X,τ), the operators such as int, cl, int cl, cl int, int cl int, cl int cl etc. are found to belong to this class of mappings. Accordingly, the weaker form of open sets including semi-open sets, Received 26 May 2015 – Accepted 25 March 2016 http://dx.doi.org/10.4995/agt.2016.3919 A. Gupta and R. D. Sarma pre-open sets, α-open sets, β-open sets are nothing but γ-open sets for different γ’s. All these families form “generalized topologies” on X. In [4], Császàr has formulated separation axioms for such spaces. Accordingly, separation axioms using semi-open sets[5], β-open sets[13], etc. become particular cases in [4]. In the same spirit, we introduce and investigate a generalized form of nor- mality called (λ,µ)-normality for generalized topologies in this paper. However, unlike in [4], we use two GT’s simultaneously in our definition. This gives us a more general definition of normality, yet it covers almost all the relevant variants of normality existing in the literature. For example, if X has a topol- ogy, then by taking λ = µ = int, we get normality for X; λ = int, µ = cl∗θ give θ-normality; λ = int, µ = cl∗δ give ∆-normality for X. If (X,τ1,τ2) is a bitopological space, then λ = intτ1 , µ = intτ2 gives rise to pairwise normality of (X,τ1,τ2). Thus our study provides a uniform approach towards various notions of normality existing in the literature. We have shown that the two most important results on normality- the Urysohn’s lemma and Tietze exten- sion theorem are valid for (λ,µ)-normality, although in a milder form. We have also defined and studied (λ,µ)-regularity in the process and provided its characterization. 2. Preliminaries Á. Császàr has defined a generalized topological space [3] in the following way: Definition 2.1. A collection G of subsets of X is called a generalized topology (in brief GT ) [3] on X if (i) ∅ ∈G; (ii) Gi ∈G for i ∈ I 6= ∅, implies G = ⋃ i∈I Gi ∈G. The same has been defined and studied as semi topological spaces by Peleg[14]. For a topological space (X,τ), each family of semi-open sets, α-open sets, pre- open sets and β-open sets etc. form a generalized topology on X. Á. Császàr [2] has used a map γ : P(X) −→ P(X) where P(X) is the power set of X, as his main tool for developing a generalized form of topological spaces. The map γ possesses the property of monotonicity, which says that, if A ⊆ B then γ(A) ⊆ γ(B). The collection of all such mappings on X is denoted by Γ(X), or simply by Γ. Definition 2.2 ([2]). Consider a non empty set X and a map γ ∈ Γ(X). We say that a subset A of X is γ-open if A ⊆ γ(A). For a topological space (X,τ), an open set (resp. semi-open, α-open, β-open, pre-open) is γ-open for γ = int (resp. cl int, int cl int, cl int cl, int cl). Also for each γ ∈ Γ(X), it may be verified that the γ-open sets form a generalized topology on X. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 8 A unified approach to normality Definition 2.3 ([2]). Let A be a subset of X and γ be a monotonic mapping on X. Then the union of all γ-open sets contained in A is called the γ-interior of A, and is denoted by iγ(A). Proposition 2.4 ([2]). A subset A of X is γ-open if and only if A = iγ(A) if and only if A is iγ-open. Definition 2.5 ([2]). A subset A of X is called γ-closed if X�A is γ-open. Definition 2.6 ([2]). The intersection of all γ-closed sets containing A is called γ-closure of A and is denoted by cγ(A). It can be shown that cγ(A) is the smallest γ-closed set containing A. Another operator called γ∗ is defined, with the help of γ in the following way: Definition 2.7 ([2]). For any A ⊆ X and γ ∈ Γ(X), we define γ∗(A) = X�(γ(X�A)) Proposition 2.8 ([2]). If γ ∈ Γ, then γ∗ ∈ Γ. Proposition 2.9 ([2]). A subset A of X is γ∗-closed if and only if γ(A) ⊆ A. Definition 2.10 ([19]). Let X be a topological space and let A ⊆ X. A point x ∈ X is in θ-closure of A if every closed neighbourhood of x intersects A. The θ-closure of A is denoted by clθ(A). The set A is called θ-closed if A = clθA. The complement of a θ-closed set is called θ-open set. Definition 2.11 ([19]). Let X be a topological space and let A ⊆ X. A point x ∈ X is in δ-closure of A if every regular open neighbourhood of x intersects A. The δ-closure of A is denoted by clδ(A). The set A is called δ-closed if A = clδ(A). The complement of a δ-closed set is called δ-open set. Definition 2.12. A topological space X is said to be (1) [8] θ-normal if every pair of disjoint closed sets one of which is θ-closed are contained in disjoint open sets; (2) [8] Weakly θ-normal if every pair of disjoint θ-closed sets are contained in disjoint open sets; (3) [6] ∆-normal if every pair of disjoint closed sets one of which is δ-closed are contained in disjoint open sets; (4) [6] Weakly ∆-normal if every pair of disjoint δ-closed sets are contained in disjoint open sets. Definition 2.13 ([9]). A bitopological space (X,τ1,τ2) is said to be pairwise normal if given a τ1-closed set A and a τ2-closed set B with A∩B = ∅, there exist τ2-open set O2 and τ1-open set O1 such that A ⊆ O2, B ⊆ O1, and O1 ∩O2 = ∅. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 9 A. Gupta and R. D. Sarma 3. (λ,µ)-regularity and (λ,µ)-normality For defining (λ,µ)-regularity and (λ,µ)-normality, no topology is required on X. It is because, for a non-empty set X, Γ(X) is also non-empty. However, we may call X a space, once we define some topological property on X, such as (λ,µ)-regularity, (λ,µ)-normality etc. Definition 3.1. Let X be a non-empty set and λ,µ ∈ Γ(X). Then X is said to be λ-regular with respect to µ if for each point x ∈ X and each λ-closed set P such that x /∈ P , there exist a λ-open set U and a µ-open set V such that x ∈ U, P ⊆ V and U ∩V = ∅. X is said to be (λ,µ)-regular if X is λ-regular with respect to µ and vice versa. Definition 3.2. A non-empty set X is called (λ,µ)-normal if for a given λ- closed set A and a µ-closed set B with A∩B = ∅, there exist a µ-open set U and a λ-open set V such that A ⊆ U, B ⊆ V and U ∩V = ∅. Below we provide characterizations for (λ,µ)-regularity and (λ,µ)-normality. Theorem 3.3. Let X be a non-empty set. Then X is (λ,µ)-regular if and only if (i) for a given x ∈ X and λ-open neighbourhood U of x, there exists a µ- closed neighbourhood V of x (that is, x ∈ V0 ⊆ V , for some V0 ⊆ X, where V0 is λ-open and V is µ-closed) such that x ∈ V ⊆ U. and (ii) for a given y ∈ X and µ-neighbourhood P of y, there exists a λ-closed neighbourhood Q of y such that y ∈ Q ⊆ P . Proof. Let x ∈ X and U be a λ-open neighbourhood of x. Therefore x /∈ X�U, a λ-closed set. Thus, there exists a disjoint pair of λ-open set O and µ-open set W such that x ∈ O, X�U ⊆ W and O ∩ W = ∅. That is, X�W ⊆ U. Hence x ∈ O ⊆ X�W ⊆ U, that is, x ∈ V ⊆ U, where V = X�W , a µ-closed set. Similarly, if X is µ-regular with respect to λ, then for a given point x ∈ X and a µ-open neighbourhood U of x, there exists a λ-closed neighbourhood V of x such that x ∈ V ⊆ U. Conversely, let x ∈ X and F be a λ-closed set such that x /∈ F. Then x ∈ X�F and X�F is λ-open. Hence by (i), there exists a λ-open set V0 and a µ-closed set V such that x ∈ V0 ⊆ V ⊆ X�F. Then by (ii), there exists a λ-open set Q0 and a µ-closed set Q such that x ∈ Q0 ⊆ Q ⊆ V0. Thus we have, x ∈ Q0, F ⊆ P0, where P0 = X�V , such that Q0 is λ-open, P0 is µ-open and P0 ∩Q0 = ∅. Hence X is (λ,µ)-regular. � Theorem 3.4. Let X be a non-empty set. Then X is (λ,µ)-normal if and only if for a given µ-closed set C and a λ-open set D such that C ⊆ D, there are a λ-open set G and a µ-closed set F such that C ⊆ G ⊆ F ⊆ D. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 10 A unified approach to normality Proof. Let C and D be the µ-closed and λ-open sets respectively such that C ⊆ D. Then X�D is a λ-closed set such that C ∩ (X�D) = ∅. Then, from the (λ,µ)-normality, there exist a λ-open set G and a µ-open set V such that C ⊆ G, X�D ⊆ V and G ∩ V = ∅. Therefore X�V ⊆ D and hence C ⊆ G ⊆ X�V ⊆ D, where X�V is a µ-closed set. Hence C ⊆ G ⊆ F ⊆ D, where X�V = F(say). Conversely, consider C as D are λ-closed and µ-closed sets respectively such that C ∩ D = ∅. Then X�C is λ-open set containing D. Then by the given hypothesis, there exist a λ-open set G and a µ-closed set F such that D ⊆ G ⊆ F ⊆ X�C. Thus, we have D ⊆ G, C ⊆ V and G∩V = ∅, where V = X�F , µ-open set. Hence X is (λ,µ)-normal. � In our next section, we provide generalized versions of Uryshon’s lemma and Tietze extension theorem, which holds for (λ,µ)-normality. 4. Uryshon’s lemma and Tietze extension Theorem for (λ,µ)-normality Definition 4.1 ([18]). Let (X,λ) be a generalized topological space and R be the real line with the usual topology. A mapping f : X → R is said to be generalized upper semi-continuous or g.u.s.c. in brief (resp. generalized lower semi-continuous or g.l.s.c. in brief) if for each a ∈ R, the set {x ∈ X : f(x) < a} (resp. {x ∈ X : f(x) > a}) is λ-open. Unlike in topology, a mapping which is both generalized upper semi-continuous and generalized lower semi-continuous, may fail to be generalized continuous in generalized topology. Example 4.2. Let X = [0, 1], λ consist of the unions of the members of the type [0,a), (b, 1], a,b ∈ [0, 1]. Let Y = X, under the usual subspace topology of R. Then the identity mapping I : X → Y is g.u.s.c. and g.l.s.c. but not generalized continuous as f−1(a,b) /∈ λ. Theorem 4.3. Let X be a (λ,µ)-normal space. Then for any disjoint pair of λ-closed set H and µ-closed set F , there exists a real valued function g on X such that (i) g(x) = 0 for x ∈ F , g(x) = 1 for x ∈ H, 0 ≤ g(x) ≤ 1, for all x ∈ X; (ii) g is λ-upper semi-continuous and µ-lower semi-continuous. Proof. Let X be a (λ,µ)-normal space and G and H be two disjoint subsets of X such that G is µ-closed and H is λ-closed. Let us consider, G0 = G and K1 = X�H. Then G0 is µ-closed and K1 is λ-open set such that G0 ⊆ K1. Since X is (λ,µ)-normal, therefore there exist a λ-open set K1/2 and a µ-closed set G1/2 such that G0 ⊆ K1/2 ⊆ G1/2 ⊆ K1. Again applying the hypothesis to each pair of sets (G0 and K1/2) and (G1/2 and K1), we obtain λ-open sets K1/4,K3/4 and µ-closed sets G1/4,G3/4 such that c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 11 A. Gupta and R. D. Sarma G0 ⊆ K1/4 ⊆ G1/4 ⊆ K1/2 ⊆ G1/2 ⊆ K3/4 ⊆ G3/4 ⊆ K1. Continuing this process, we obtain two families {Gi} and {Ki}, where i = p/2q, where {p = 1, 2, . . . , 2q − 1,q = 1, 2, . . .}. If i is any other dyadic rational number other than p/2q, then let Ki = ∅, whenever i ≤ 0 and Ki = X, for i > 1. Similarly, Gi = ∅ for i < 0 and Gi = X for i ≥ 1. Thus, for every r ≤ s ≤ t, we have Kr ⊆ Ks ⊆ Gs ⊆ Gt and for s < t, we have Gs ⊆ Kt. Now, we define a function g : X → [0, 1] such that g(x) = inf{t | x ∈ Kt} Clearly, g(x) ∈ [0, 1]. If x ∈ G, then x ∈ Ki for all i, therefore g(x) = 0, when x ∈ H = X�K, then x /∈ Ki for all i ∈ [0, 1], hence g(x) = 1. Now, we have to show that g is λ-upper semi-continuous and µ-lower semi- continuous. First we show that (i) if x ∈ Gp then g(x) ≤ p (ii) if x /∈ Kp, then g(x) ≥ p. Let x ∈ Gp, then x ∈ Ks for every s > p. Therefore g(x) ≤ p. Similarly, if x /∈ Kp, then x /∈ Ks for any s < p, hence g(x) ≥ p. Thus we can say that whenever g(x) > p, we have x /∈ Gp and g(x) < p, we have x ∈ Kp. Now, we consider, x ∈ g−1([0,a)), then g(x) ∈ [0,a), that is, there exists t < a such that g(x) < t and hence x ∈ Kt, therefore g−1([0,a)) ⊆ ⋃ t t and hence x /∈ Gt, that is, x ∈ X�Gt, therefore g−1((a, 1]) ⊆ ⋃ t>a (X�Gt). Conversely, let x ∈ ⋃ t>a (X�Gt), then x ∈ X�Gi for some i > a, thus g(x) ≥ i > a and x ∈ g−1((a, 1]). Therefore g−1((a, 1]) = ⋃ t>a (X�Gt), a µ-open set. Hence g is µ-lower semi-continuous function. � Our next theorem resembles with the classical Tietze extension theorem. But before that, we quote a result which will be used in our main theorem. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 12 A unified approach to normality Theorem 4.4 ([3]). Let (X,λ) be a generalized topological space and (Y,λy) be a generalized topological subspace of X. Then a subset A of Y is λ-closed in Y if and only if it is the intersection of Y with a λ-closed set in X. Now we come to our main proposed result. Theorem 4.5. Let X be a (λ,µ)-normal space. Let A ⊆ X be a λ-closed as well as µ-closed set and f be a real valued function defined on A which is λ-upper semi-continuous as well as µ-lower semi-continuous function. Then there exists an extension F of f to the whole of X such that F is λ-upper semi-continuous and µ-lower semi-continuous in X. Proof. Let X be a (λ,µ)-normal space and A be a λ-closed and µ-closed subset of X. Suppose f is a real valued function on A which is λ-upper semi-continuous and µ-lower semi-continuous. Let n be a positive integer, then for each integer k, let Unk = {x : f(x) ≥ k/n} and L n k = {x : f(x) ≤ (k − 1)/n} Then, for every integer k, Unk and L n k are λ-closed and µ-closed subsets of A respectively. Therefore Unk and L n k are λ-closed and µ-closed subsets of X also and Unk ∩L n k = ∅. By Theorem 4.3, for each k = 0, 1, 2, . . ., there is a function Uk defined on X, which is λ-upper semi-continuous and µ-lower semi-continuous, such that Uk(x) = 0, for x ∈ Lnk Uk(x) = 1/n, for x ∈ U n k and 0 ≤Uk(x) ≤ 1/n for all x ∈ X Also, for each k = 0,−1,−2, . . ., there is a function Vk defined on X which is λ-upper semi-continuous and µ-lower semi-continuous function, such that Vk(x) = −1/n, for x ∈ Lnk Vk(x) = 0, for x ∈ U n k and −1/n ≤Vk(x) ≤ 0 for all x ∈ X Now, we know that Lnk ⊆ L n k+1 and U n k+1 ⊆ U n k . Thus for k = 0,−1,−2, . . ., if f(x) ≥ 0, then x ∈ Unk and hence Vk(x) = 0. Similarly, for k = 1, 2, . . ., if f(x) ≤ 0, then x ∈ Lnk and hence Uk(x) = 0. Now, we define a real valued function fn on X as follows: fn(x) = ∞∑ k=1 Uk(x) + 0∑ k=−∞ Vk(x) for x ∈ X Since Ui and Vi are λ-upper semi continuous and µ-lower semi-continuous func- tion therefore fn is also a λ-upper semi continuous and µ-lower semi-continuous function. Now, consider Cnk = U n k−1 ∩L n k+1 for k ∈ Z. Thus C n k ⊆ U n k−1 and C n k ⊆ L n k+1, that is, Cnk = {x : k−1 n ≤ f(x) ≤ k n }. Therefore A = ⋃ {Cnk ,k ∈ Z}. Now, let k ≥ 1, and x ∈ Cnk , that is, 0 ≤ f(x) ≤ k/n. Since x ∈ L n k+1, therefore Uj(x) = 0 for j ≥ k + 1 and x ∈ Unk−1, therefore Uj(x) = 1/n for 1 ≤ j ≤ k−1. Also 0 ≤Uk(x) ≤ 1/n. Thus c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 13 A. Gupta and R. D. Sarma fn(x) = U1(x) + U2(x) + . . . + Uk−1(x) + Uk(x) + Uk+1(x) + . . . fn(x) = k−1 n + Uk(x) Therefore |f(x) −fn(x)| ≤ |k/n− (k − 1)/n−Uk(x)| ≤ 1/n + |Uk(x)| ≤ 2/n, for x ∈ Cnk . Now, let k ≤ 0, and x ∈ Cnk , that is, −(k + 1)/n ≤ f(x) ≤ 0. Since x ∈ L n k+1, therefore Vj(x) = −1/n for (k − 1) ≤ j ≤ 0 and x ∈ Unk−1, therefore Vj(x) = 0 for j ≤ (k + 1). Thus fn(x) = V0(x) + V−1(x) + V−2(x) + . . .Vk−1(x) + Vk(x) + Vk+1(x) + . . . fn(x) = k n + Vk(x), for a non positive integer k Therefore |f(x) −fn(x)| ≤ 2/n Hence |f(x) −fn(x)| ≤ 2/n for all x ∈ A. We recall that g-nets defined in [17] behave almost the same way in generalized topology as the nets do in topology and a sequence is just a particular case of g-nets in generalized topology. Due to this fact, fn|A for n = 1, 2, . . . converge uniformly to f on A. Also (fn|A) forms a Cauchy sequence with respect to the uniform norm on A. As a result, as in[9], f has an extension F to X. It may be easily shown that F is λ-upper semi-continuous and µ-lower semi-continuous function. � 5. Conclusion Under different set of conditions, we get different variants of normality. If we take (i) λ = µ = interior operator of a topology on X, then the λ-closed sets and µ-closed sets are nothing but the closed sets of X. Therefore (λ,µ)- normality just becomes normality. (ii) λ = intτ1 and µ = intτ2 , two different interior operators over two different topologies τ1 and τ2, then (λ,µ)-normality becomes pairwise normality [9] of (X,τ1,τ2) (iii) λ =interior operator and µ = cl∗θ operator, then (λ,µ)-normality becomes θ-normality[8]. This is due to fact that clθ ∈ Γ, that is, clθ operator is monotonic. First we verify that for A,B ⊆ X such that A ⊆ B, we have clθ(A) ⊆ clθ(B). Let x ∈ clθ(A), then every closed neighbourhood of x intersects A. Since A ⊆ B, therefore every closed neighbourhood of x intersects B also. Hence x ∈ clθ(B). Thus clθ(A) ⊆ clθ(B). Therefore clθ ∈ Γ. Hence by Proposition 2.8, cl∗θ ∈ Γ. Now, let A be µ-closed, that is, cl∗θ-closed set. Then by Proposition 2.9 clθ(A) = (cl ∗ θ) ∗(A) and by Proposition 2.8 clθ(A) = (cl ∗ θ) ∗(A) ⊆ A, that is, A is θ-closed. Since every θ-open set is open therefore in the light of (λ,µ)-normality, we have disjoint pair of sets in which one is λ-closed, that is, closed set and the other is µ-closed, that is, θ-closed set separated c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 14 A unified approach to normality by disjoint µ-open, that is, θ-open set and hence open set and λ-open set, that is, open set respectively. (iv) λ =interior operator and µ = cl∗δ operator, then (λ,µ)-normality becomes ∆-normality [6]. Because clδ ∈ Γ, that is, clδ operator is monotonic. As, consider A,B ⊆ X such that A ⊆ B. Then clδ(A) ⊆ clδ(B). Since x ∈ clδ(A), then every regular open neighbourhood of x intersects A. Since A ⊆ B, therefore every regular open neighbourhood of x intersects B also. Hence x ∈ clδ(B). Thus clδ(A) ⊆ clδ(B). Therefore clδ ∈ Γ. From the Proposition 2.8 cl∗δ ∈ Γ also. 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