CoUzViAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 9, No. 1, 2008 pp. 39-50 Low separation axioms via the diagonal Maŕıa Luisa Colasante, Carlos Uzcátegui and Jorge Vielma∗ Abstract. In the context of a generalized topology g on a set X, we give in this article characterizations of some separation axioms between T0 and T2 in terms of properties of the diagonal in X × X. 2000 AMS Classification: 54A05, 54D10. Keywords: Generalized topologies, intersection structures, envelope opera- tions, kerneled and saturated sets. 1. Introduction A well known elementary fact says that a topological space X is Hausdorff iff the diagonal ∆ is closed in X × X. In this paper we show that behind this observation there is a general pattern which includes several separation axioms below T2 (namely T0, T1/4, T1/2, T1, R0 and R1). These low separation axioms have been studied in a more general setting where, instead of open sets, other kind of subsets are used: semi-open sets, α-open sets, λ-open sets, etc. ([1], [6], [8]). These families (called generalized topologies in [5]) always contain ∅ and X and are closed under arbitrary unions (but not necessarily under finite intersections). On the other hand, in the study of low separation axioms, set operations similar to the closure operator are frequently used. These operations are naturally extended to the context of a generalized topology g (and are then called envelope operations [5]). For instance, kg(A) corresponds to the topological closure of A, χg(A) corresponds to the kernel of A [7] (i.e. the intersection of all open sets containing A) and satg(A) corresponds to the union of the closure of points in A. Our characterizations of the separation axioms are in terms of the behavior of ∆ under kg, χg and satg. An example of our results is that g satisfies T1 iff χg(∆) = ∆. We will also give a characterization of low separation axioms in terms of saturated sets. A set in a topological space is said to be saturated when it ∗The authors would like to thank the partial support provided by the Universidad de Los Andes CDCHT grant A-1335-05-05. 40 M. L. Colasante, C. Uzcátegui and J. Vielma contains the closure of each of its points. It is known that a topology satisfies the axiom R0 iff every open set is saturated [5]. Another notion of saturation was studied in [4]. We extend these notions and study its connection with low separation axioms. The paper is organized as follows. In section 2 we recall the basic separation axioms and state some facts about generalized topologies and envelope oper- ations. The results about the properties of the diagonal and the separation axioms are shown in section 3. In section 4 we study the family of saturated sets and its connection with the separation axioms. Finally, in section 5 we analyze the axioms T1/2 and T1/4. 2. Preliminaries We follow the notations and definitions used in [5]. A subset g of the power set P(X) of a set X is a generalized topology (briefly GT ) on X if {∅, X} ⊆ g and g is closed under arbitrary unions. If g is a generalized topology, then the family of complements of sets in g is usually called an intersection structure. In this article g will always denote a generalized topology. Definition 2.1 ([5]). An envelope operation on X is a mapping ρ : P(X) → P(X) such that (i) A ⊆ ρA for A ⊆ X. (ii) If A ⊆ B, then ρA ⊆ ρB for all A ⊆ B ⊆ X. (iii) ρA = ρρA for A ⊆ X. More generally, ρ : P(X) → P(X) is called a weak envelope if (i) and (ii) are satisfied. Examples of envelope operations are given below. Definition 2.2. Let A ⊆ X. (i) χg(A) = ⋂ {H ∈ g : A ⊆ H}. (ii) kg(A) = {x ∈ X : K ∩ A 6= ∅ for each K ∈ g with x ∈ K}. (iii) satg(A) = ⋃ x∈A kg({x}). The operator χg and kg were defined in [5] and shown to be envelope oper- ations. It is straightforward to show that satg is an envelope. It is also easy to see that χg(A) = A for all A ∈ g. Moreover, x ∈ χg(y) if and only if y ∈ kg(x) for any x, y ∈ X (where we write ρ(x) instead of ρ({x}) for any set operator ρ). When g is a topology, kg is the closure operator cl and χg(A) is the kernel of A, frequently denoted by  or Ker(A). Notice that, in general, if τ is the topology generated by a GT g, then kg 6= clτ and χg 6= Kerτ (for instance, in R take g to be the GT generated by the collection of intervals of the form (−∞, a) and (a, +∞)). Now we formulate the fundamental separation axioms in terms of an arbi- trary GT ([5]). (T0) For all x, y ∈ X, with x 6= y there is K ∈ g containing precisely one of x and y. Low separation axioms via the diagonal 41 (T1) For all x, y ∈ X with x 6= y there is K ∈ g such that x ∈ K, y /∈ K. (T2) For all x, y ∈ X, x 6= y, there are K, K ′ ∈ g such that x ∈ K, y ∈ K′ and K ∩ K′ = ∅. (R0) For all x, y ∈ X, if kg(x) 6= kg(y), then kg(x) ∩ kg(y) = ∅. (R1) For all x, y ∈ X, if kg(x) 6= kg(y), then there are K, K ′ ∈ g disjoint such that kg(x) ⊆ K and kg(y) ⊆ K ′. Proposition 2.3 ([5]). g satisfies (R0) iff for all x, y ∈ X, if there is K ∈ g such that x ∈ K and y 6∈ K, then there is K′ ∈ g such that x 6∈ K′ and y ∈ K′. Two of the recently widely studied separation axioms below T1 can be stated for a generalized topology g as follows: (T1/2) For all x ∈ X, {x} ∈ g or {x} = kg(x). (T1/4) For all x ∈ X, {x} = χg(x) or {x} = kg(x). In the rest of this section we introduce some notions and present some basic facts about the envelope operations χg, kg and satg that will be used in the sequel. In order to simplify the notation, we will write χ(A), k(A) and sat(A) avoiding the use of g. We say that a set A is closed (resp. kerneled) iff k(A) = A (resp. χ(A) = A). When g is a topology, kerneled sets are usually called Λ-sets [7]. A subset A ⊆ X is said g-saturated (or just saturated) if sat(A) = A, equivalently if k(x) ⊆ A for all x ∈ A. Note that sat(A) is the smallest saturated set containing A, and that A is saturated if and only if X\A is kerneled, where X\A denotes the complement of A. In particular sat(x) = k(x), for any x ∈ X. The collection of all saturated subsets of X is denoted by S(X). It is easy to see that S(X) is closed under arbitrary unions and arbitrary intersections. Proposition 2.4 ([5]). A is closed iff X\A ∈ g. Proof. Let A closed. If y ∈ X\A then y /∈ k(A), and hence there exists By ∈ g such that y ∈ By and By ∩A = ∅. Thus X\A = ⋃ y∈X\A By ∈ g. The converse is obvious. � Since our analysis of the separation axioms will be in terms of the behavior of the diagonal ∆, we need to introduce the GT on the product X × X. Let g be a GT on X, then the family g2 below is a GT in X2: g 2 = { D ⊆ X × X : D = ⋃ α Aα × Bα, with Aα, Bα ∈ g } . In this article the operators k, χ and sat on X × X refer to the generalized topology g2. Proposition 2.5. For any (x, y) ∈ X × X the following holds: (i) χ(x, y) = χ(x) × χ(y). (ii) k(x, y) = k(x) × k(y). Proof. (i) Let (p, q) ∈ χ(x, y) and A, B ∈ g with x ∈ A and y ∈ B. Then (x, y) ∈ D = A × B ∈ g2 and so (p, q) ∈ A × B. Thus p ∈ χ(x) and q ∈ χ(y). 42 M. L. Colasante, C. Uzcátegui and J. Vielma Conversely, if (p, q) ∈ χ(x) × χ(y) and D ∈ g2, then (x, y) ∈ D = ⋃ α Aα × Bα, with Aα, Bα ∈ g. There is α such that x ∈ Aα and y ∈ Bα and hence (p, q) ∈ Aα × Bα ⊆ D. This implies that (p, q) ∈ χ(x, y). � Proposition 2.6. (i) If A = ⋃ i∈I Ai, then χ(A) = ⋃ i∈I χ(Ai). (ii) If A = ⋃ i∈I Ai, then sat(A) = ⋃ i∈I sat(Ai). Proof. (i) was proved in [5] and (ii) is obvious. � Proposition 2.7. Let A be a subset of X × X. Then (i) χ(A) = ⋃ (x,y)∈A χ(x) × χ(y). (ii) sat(A) = ⋃ (x,y)∈A k(x) × k(y). Proof. The result follows directly from propositions 2.5 and 2.6. � To end this section, we introduce three more operations. Let A ⊆ X, then define kθ(A) = {x ∈ X : A ∩ k(D) 6= ∅ for each D ∈ g such that x ∈ D} kλ(A) = k(A) ∩ χ(A) kµ(A) = sat(A) ∩ χ(A) It is easy to see that kθ is a weak envelope on X such that k(A) ⊆ kθ(A) for all A ⊆ X. Also, it is straightforward to show that kλ and kµ are envelope operations on X (more generally, the finite intersection of envelope operations is again an envelope). When g is a topology, kθ is the well known clθ operator [9, 4] and kλ is the clλ operator [2]. The clλ-closed sets (i.e. sets such that clλ(A) = A) are usually called λ-closed sets and their complements λ-open sets [1]. If g is the GT consisting of the λ-open sets, then k = clλ. 3. Separation axioms as properties of the diagonal We will denote by ∆ the diagonal in X × X. In this section we show that the separation axioms can be characterized in terms of χ(∆), sat(∆) and k(∆). Besides ∆ there are two others binary relations which play an important role in what follows. (x, y) ∈ Lg iff ∀A ∈ g [x ∈ A → y ∈ A] (x, y) ∈ Eg iff ∀A ∈ g [x ∈ A ↔ y ∈ A] . Notice that ∆ ⊆ Eg ⊆ Lg. Moreover, Lg is transitive relation and Eg is an equivalence relation on X. The main result of this section is summarized in the following table. T2 ⇔ k(∆) = ∆ T1 ⇔ χ(∆) = ∆ ⇔ sat(Eg) = ∆ T0 ⇔ ∆ = Eg R0 ⇔ χ(∆) = Eg ⇔ Eg = sat(∆) R1 ⇔ k(∆) = Eg In order to show these results we need several auxiliary lemmas. Low separation axioms via the diagonal 43 Lemma 3.1. (i) (x, y) ∈ Lg iff y ∈ χ(x) iff x ∈ k(y). (ii) (x, y) ∈ Eg iff k(x) = k(y) iff χ(x) = χ(y). Proof. Since χ(x) = ∩{A ∈ g : x ∈ A}, (i) follows directly from the definition of Lg. Part (ii) follows from the symmetry of the relation Eg. � The following result characterizes χ, k, and sat for the diagonal ∆ on X × X. In particular, it shows that k(∆), χ(∆) and sat(∆) are symmetric (and obviously reflexive) relations. Lemma 3.2. (i) (x, y) ∈ k(∆) iff A ∩ B 6= ∅ for all A, B ∈ g such that x ∈ A and y ∈ B. (ii) (x, y) ∈ χ(∆) iff k(x) ∩ k(y) 6= ∅. (iii) (x, y) ∈ sat(∆) iff χ(x) ∩ χ(y) 6= ∅. Proof. (i) Let (x, y) ∈ k(∆) and let A, B ∈ g with x ∈ A and y ∈ B. Then (x, y) ∈ A × B ∈ g2 and thus A × B ∩ ∆ 6= ∅, which implies A ∩ B 6= ∅. Reciprocally, let D ∈ g2 containing (x, y). Then D = ⋃ α Aα × Bα, with Aα, Bα ∈ g. It follows that (x, y) ∈ Aα × Bα for some α. By assumption Aα ∩ Bα 6= ∅. If z ∈ Aα ∩ Bα, then (z, z) ∈ D ∩ ∆. Therefore (x, y) ∈ k(∆). (ii) (x, y) ∈ χ(∆) = ⋃ x∈X χ(x) × χ(x) if and only if there is z ∈ X such that (x, y) ∈ χ(z) × χ(z) for some z ∈ X if and only if (z, z) ∈ k(x) × k(y). (iii) Follows by a similar argument as (ii). � From lemma 3.2(i), it follows that (x, y) ∈ k(∆) iff ∀A ∈ g [x ∈ A → y ∈ k(A)] iff ∀B ∈ g [y ∈ B → x ∈ k(B)]. Therefore we have the following fact about the operator kθ (defined in section 2). Lemma 3.3. (x, y) ∈ k(∆) iff y ∈ kθ(x) iff x ∈ kθ(y). We prove that the envelope operations k, χ and sat coincide on the relations ∆, Lg and Eg. Lemma 3.4. (i) k(∆) = k(Lg) = k(Eg). (ii) χ(∆) = χ(Lg) = χ(Eg). (iii) sat(∆) = sat(Lg) = sat(Eg). Proof. (i). Since ∆ ⊆ Eg ⊆ Lg, it suffices to show that Lg ⊆ k(∆). Let (x, y) ∈ Lg and let A, B ∈ g with (x, y) ∈ A × B. By lemma 3.1, x ∈ k(y) and thus y ∈ χ(x). Since χ(x) ⊆ A then y ∈ A. It follows that (y, y) ∈ A × B. Therefore (x, y) ∈ k(∆), by definition of k(∆). (ii). As in (i) we only show that Lg ⊆ χ(∆). Let (x, y) ∈ Lg. By proposition 2.7, χ(∆) = ⋃ z∈X χ(z) × χ(z). Thus, if (x, y) /∈ χ(∆), then in particular y /∈ χ(x) and this implies that there is A ∈ g containing x such that y /∈ A, a contradiction. (iii). If (x, y) ∈ Lg then, by lemma 3.1(i) and proposition 2.7, (x, y) ∈ k(y) × k(y) ⊆ sat(∆). Hence Lg ⊆ sat(∆). � 44 M. L. Colasante, C. Uzcátegui and J. Vielma Proposition 3.5. (i) g satisfies (T2) iff kθ(x) = {x} for each x ∈ X. (ii) g satisfies (T1) iff k(x) = {x} for each x ∈ X iff χ(x) = {x} for each x ∈ X. (iii) g satisfies (T0) iff kλ(x) = {x} for each x ∈ X. That is to say, k(x) ∩ χ(x) = {x} for each x ∈ X. Proof. (i) First note that, if A, B ∈ g, then A ∩ B = ∅ iff A ∩ k(B) = ∅. Suppose g satisfies (T2). Given x ∈ X and y 6= x, there exist A, B ∈ g such that x ∈ A, y ∈ B and A∩B = ∅, then y /∈ kθ(A) and, in particular, y /∈ kθ(x). Therefore kθ(x) = {x} for each x ∈ X. Conversely, suppose kθ(x) = {x} for each x ∈ X. Given x, y ∈ X, if x 6= y then y /∈ kθ(x). Thus (x, y) /∈ k(∆) and hence, by lemma 3.2, there exist A, B ∈ g such that x ∈ A, y ∈ B and A ∩ B = ∅, which shows that g satisfies (T2). (ii) g satisfies (T1) iff given x ∈ X and y 6= x, there exist A ∈ g such that x ∈ A and y /∈ A, iff given x ∈ X and y 6= x, y /∈ k(x), iff k(x) = {x} for each x ∈ X. For the second part, note that if k(x) = {x} for each x ∈ X, then the set X\{x} = ⋃ y 6=x k(y) is saturated for each x ∈ X, and thus {x} is kerneled for each x ∈ X. A similar argument shows the reverse implication. (iii) g satisfies (T0) iff given x ∈ X and y 6= x, y /∈ k(x) or x /∈ k(y), iff y /∈ k(x) or y /∈ χ(x), iff kλ(x) = k(x) ∩ χ(x) = {x} for each x ∈ X. � Now we start showing the main results of this section. Theorem 3.6. (i) g satisfies (T2) iff k(∆) = ∆ iff ∆ is closed. (ii) g satisfies (T1) iff χ(∆) = ∆ iff Lg = ∆ iff ∆ is saturated iff ∆ is kerneled. (iii) g satisfies (T0) iff Eg = ∆. Proof. (i) By proposition 3.5(i), g satisfies (T2) iff kθ(x) = {x} for each x ∈ X, iff y 6= x implies y /∈ kθ(x) iff (x, y) /∈ k(∆). The second part is obvious. (ii) Suppose g satisfies (T1). From proposition 3.5(ii), k(x) = {x} for each x ∈ X. If (x, y) ∈ χ(∆), then k(x) ∩ k(y) 6= ∅ and thus x = y. On the other hand, if χ(∆) = ∆ and x 6= y, then (x, y) /∈ χ(∆) and thus k(x) ∩ k(y) = ∅. In particular x /∈ k(y), so there exists A ∈ g such that x ∈ A and y /∈ A. Therefore g satisfies (T1). The second and third parts follow from lemma 3.1(i) and proposition 3.5(ii). The last part is obvious. (iii) g satisfies (T0) iff for all x 6= y, y /∈ χ(x) or x /∈ χ(y) iff χ(x) 6= χ(y) iff (x, y) /∈ Eg iff Eg = ∆. � Theorem 3.7. g satisfies (R0) iff χ(∆) = Eg iff χ(∆) = Lg iff Eg is kerneled iff Eg is saturated. Proof. Since Eg ⊆ χ(Eg) = χ(∆), then χ(∆) = Eg iff χ(∆) ⊆ Eg. From proposition 2.3, g satisfies (R0) iff x, y ∈ X implies k(x) = k(y) or k(x)∩k(y) = ∅. Therefore g satisfies (R0) iff χ(∆) = Eg. The second part of the equivalence follows from the fact that Eg ⊆ Lg ⊆ χ(Lg) = χ(∆). The third part is obvious. Low separation axioms via the diagonal 45 On the other hand, since y ∈ k(x) iff x ∈ χ(y), then g satisfies (R0) iff the sets χ(x), x ∈ X, form a partition of X, iff sat(∆) = Eg. � Theorem 3.8. g satisfies (R1) iff k(∆) = Eg iff k(∆) = Lg iff Eg is closed. Proof. g satisfies (R1) iff x, y ∈ X, and k(x) 6= k(y), implies the existence of A, B ∈ g such that x ∈ A, y ∈ B and A ∩ B = ∅ iff (x, y) /∈ Eg implies (x, y) /∈ k(∆) iff k(∆) ⊆ Eg. Since Eg ⊆ k(Eg) = k(∆), it follows that g satisfies (R1) iff k(∆) = Eg. The other two equivalences are obvious. � Corollary 3.9. (i) g satisfies (R0) iff k(x) = χ(x) for each x ∈ X. (ii) g satisfies (R1) iff kθ(x) = χ(x) = k(x) for each x ∈ X. Proof. (i) and (ii) follow from theorems 3.7 and 3.8 respectively. � Remark 3.10. If X is a topological space, and g is the family of the λ-open sets, then kg(x) and χg(x) are usually denoted clλ(x) and λker(x) respectively [3]. These envelope operations satisfy that clλ(x) = λKer(x) for all x ∈ X. In fact, since every open set and every closed set is λ-open, then λKer(x) ⊆ cl(x)∩Ker(x) = clλ(x). On the other hand, since every λ-open set is the union of an open set and a saturated set, then clλ(x) ⊂ A for every λ-open set A containing x. From this and corollary 3.9, every topological space X is λ-R0, a fact that was unnoticed by the authors of [3]. 4. Relations, saturated sets and separation axioms In this section we will introduce the notion of a saturated set with respect to a binary relation (like k(∆), Lg and Eg). We will show that the results of the previous section can be stated in terms of algebraic properties of the collection of saturated sets. Let E be a binary relation on a set X (i.e. E ⊆ X × X). We will always assume that E contains the diagonal ∆. We say that a subset A ⊆ X is E-saturated if whenever x ∈ A and (y, x) ∈ E, then y ∈ A. The family of E-saturated sets will be denoted by S[E]. The following result shows that the notion of an E-saturated set is a natural generalization of a g-saturated set. Proposition 4.1. (i) A ∈ S[Lg] iff for each x ∈ A, k(x) ⊆ A, i.e. S(X) = S[Lg]. (ii) A ∈ S[Eg] iff kλ(x) ⊆ A, for each x ∈ A. (iii) A ∈ S[k(∆)] iff kθ(x) ⊆ A, for each x ∈ A. Proof. The proof follows from the fact that (y, x) ∈ Lg iff y ∈ k(x), (y, x) ∈ Eg iff k(x) = k(y) iff y ∈ kλ(x) = k(x) ∩ χ(x), and (y, x) ∈ k(∆) iff y ∈ kθ(x). � We show below a general fact about saturated sets which will be used several times in the sequel. 46 M. L. Colasante, C. Uzcátegui and J. Vielma Lemma 4.2. Let E be a binary relation over X. (i) S[E] is closed under arbitrary unions and intersections. (ii) If E is a symmetric relation, then S[E] is a complete atomic Boolean algebra. Moreover, S[E] = S[F ] where F is the smallest equivalence relation containing E and the F -equivalence classes are the atoms of S[E]. (iii) S[E] = P(X) iff E = ∆. Proof. (i) is obvious. (ii) To get the result it is enough to prove that S[E] is closed under complements. Let A ∈ S[E]. If X\A /∈ S[E], there exists x, y ∈ X such that y ∈ A and (y, x) ∈ E but x /∈ A. From the symmetry of E, it follows that (x, y) ∈ E which implies that x ∈ A, a contradiction. Let F be the transitive closure of E, that is to say, (x, y) ∈ F if there are xi ∈ X, i = 0, · · · , n such that x = x0, y = xn and (xi, xi+1) ∈ E. It is easy to check that F is the smallest equivalence relation containing E. Therefore S[F ] ⊆ S[E]. On the other hand, it is routine to verify that each F -equivalence class [x]F is E-saturated. Moreover, if z ∈ A ⊆ [x]F and A is E-saturated, then [z]F = [x]F and thus A = [x]F . Hence the F -equivalence classes are the atoms of S[E] and S[E] = S[F ]. (iii) One direction is obvious. For the other, suppose E is not equal to ∆ and let (x, y) ∈ E with x 6= y. Then {y} is not E-saturated. � Remark 4.3. (i) Since k(∆), χ(∆) and Eg are symmetric relations (lemma 3.2), then S[k(∆)], S[χ(∆)] and S[Eg] are complete atomic Boolean algebras. Now from theorem 3.6 and lemma 4.2, it follows immediately that g satisfies T2 iff S[k(∆)] = P(X) iff every cofinite set belongs to S[k(∆)]. Clearly the axioms T1 and T0 are characterized in an analogous way. (ii) If g is a topology, S[k(∆)] is denoted by Bθ(X) in [4]. It was proved there that Bθ(X) is complete Boolean algebra. Note that this result is an immediate consequence of lemma 4.2(ii). Our next results deal with the axioms (R0) and (R1). Theorem 4.4. The following are equivalent. (i) g satisfies (R0). (ii) S[Lg] is a complete atomic Boolean algebra. (iii) g ⊆ S[Lg]. Proof. The equivalence (i) ↔ (iii) was proved in [5] lemma 3.2. It is clear that g satisfies (R0) iff Lg is a symmetric relation. Therefore (i) → (ii) follows from lemma 4.2(ii). For the reverse implication, note that if x ∈ X and z ∈ k(x), then k(z) ⊆ k(k(x)) = k(x), thus k(x) ∈ S(X). Suppose S[Lg] is a complete Boolean algebra, and let y ∈ X and x ∈ k(y). If y /∈ k(x), then y ∈ X\k(x) ∈ S(X) and we will have that x ∈ k(y) ⊆ X\k(x), a contradiction. Thus y ∈ k(x) which shows that (ii) → (i). � Low separation axioms via the diagonal 47 Theorem 4.5. The following are equivalent. (i) g satisfies (R1). (ii) S[k(∆)] is a complete atomic Boolean algebra and the sets k(x) (x ∈ X) are its atoms. (iii) g ⊆ S[k(∆)]. Proof. (i) → (ii). Suppose g satisfies (R1). Since k(∆) is symmetric, then by lemma 4.2 S[k(∆)] is a complete atomic Boolean algebra. Since k(∆) = Lg (theorem 3.8), then each k(x) is k(∆)-saturated. To show that the sets k(x) are the atoms, let z ∈ A ⊆ k(x) with A a k(∆)-saturated set. Then z ∈ k(x) and thus (z, x) ∈ k(∆). By symmetry (x, z) ∈ k(∆) and as A is k(∆)-saturated, then x ∈ A. Hence A = k(x). (ii) → (iii). Suppose (ii) holds. We will show that S[Lg] = S[k(∆)] and the result will follow from theorem 4.4. Since Lg ⊆ k(Lg) = k(∆) (lemma 3.4), then S[k(∆)] ⊆ S[Lg]. Conversely, if A is Lg-saturated, then A is equal to the union of the sets k(x) with x ∈ A. But by hypothesis each k(x) belongs to the complete algebra S[k(∆)], thus A ∈ S[k(∆)]. (iii) → (i). Suppose g ⊆ S[k(∆)]. We will show that k(∆) = Lg, and from this and theorem 3.8 the result follows. Let (x, y) ∈ k(∆). Given A ∈ g with y ∈ A, then A ∈ S[k(∆)] and thus x ∈ A. Then x ∈ k(y) and therefore (x, y) ∈ Lg. Since Lg ⊆ k(Lg) = k(∆), we conclude that k(∆) = Lg. � 5. T1/2 and T1/4 In this section we characterize the axioms T1/2 and T1/4 in terms of proper- ties of the diagonal and also in terms of properties of the family of saturated sets. We start with a general result about envelope operations. Lemma 5.1. Let g be a generalized topology on X and let ρ be an envelope such that ρ(x) = k(x) for all x ∈ X. For each A ⊆ X, the following are equivalent: (i) A = ρ(A) ∩ χ(A). (ii) ρ(x) ⊆ ρ(A) \ A, for all x ∈ ρ(A) \ A. Proof. (i) → (ii). Suppose A = ρ(A) ∩ χ(A) and let x ∈ ρ(A) \ A. Then x /∈ χ(A) and thus there exists H ∈ g such that A ⊆ H and x /∈ H. Let y ∈ ρ(x) ⊂ ρ(A). If y ∈ A, then y ∈ H and it must be that x ∈ H since y ∈ k(x) = ρ(x), a contradiction. Thus y /∈ A and therefore ρ(x) ⊆ ρ(A) \ A. (ii) → (i). Conversely, suppose ρ(x) ⊆ ρ(A) \ A, for all x ∈ ρ(A) \ A. Let z ∈ ρ(A)∩χ(A). If z /∈ A, then ρ(z) ⊆ ρ(A)\A and it is clear that A ⊂ X \ρ(z). Since ρ(z) = k(z), z ∈ χ(A) and X \ k(z) ∈ g (proposition 2.4), then it must be that z ∈ X \ k(z), a contradiction. Therefore A = ρ(A) ∩ χ(A). � Recall from section 2 the definition of the envelope operations kλ(A) = k(A) ∩ χ(A) and kµ(A) = sat(A) ∩ χ(A), A ⊂ X. We denote A ′ = k(A) \ A and A∗ = sat(A) \ A. The following result is an immediate consequence of lemma 5.1. 48 M. L. Colasante, C. Uzcátegui and J. Vielma Corollary 5.2. Let A ⊆ X. Then (i) A = kλ(A) iff A ′ ∈ S(X). (ii) A = kµ(A) iff A ∗ ∈ S(X). It is known that a topological space X is T1/2 iff every subset of X is λ-closed [1]. This result inspired part of the theorems 5.3 and 5.4 that follows. Theorem 5.3. The following are equivalent. (i) g satisfies (T1/4). (ii) A = kµ(A) for all A ⊂ X. (iii) A∗ ∈ S(X) for all A ⊂ X. (iv) if sat(A) ⊂ χ(A), then sat(A) = A. Proof. The equivalence (ii) ↔ (iii), follows from corollary 5.2(ii). (i) ↔ (ii). Suppose g satisfies (T1/4) and let A ⊂ X. Let A1 = {x ∈ X \ A : χ(x) = {x}} and A2 = X \ (A ∪ A1). Notice that A1 is kerneled (by proposition 2.6) and A2 is saturated (since k(x) = {x} for every x ∈ A2). Since A = Ac1 ∩ A c 2, then sat(A) ⊆ A c 1 and χ(A) ⊆ A c 2. Therefore A = sat(A) ∩ χ(A). Conversely, suppose that A = sat(A)∩χ(A) for all A ⊂ X and let x ∈ X. If {x} is not kerneled, then X \ {x} is not saturated. Since X is the only saturated set containing X \ {x}, this set must be kerneled, and thus {x} is saturated. (iv) ↔ (i). Suppose (iv) holds and let x ∈ X. If {x} is not kerneled, then X \ {x} is not saturated. Hence there exists y ∈ sat(X \ {x}) such that y /∈ χ(X \ {x}), which implies that X \ {x} is kerneled and therefore {x} is closed. Conversely, suppose (i) holds and let A ⊂ X such that sat(A) ⊂ χ(A). If A is not saturated, there is x ∈ sat(A) \ A. By hypothesis {x} is kerneled or closed. If {x} is kerneled, then X \ {x} is a saturated set containing A, thus X \ {x} contains sat(A), a contradiction. If {x} is closed, then X \ {x} is kerneled and hence X \{x} ⊃ χ(A) ⊃ sat(A), again a contradiction. Therefore sat(A) = A. � By replacing the envelope sat by the envelope k in the proof of theorem 5.3, we obtain the following result. Theorem 5.4. The following are equivalent. (i) g satisfies (T1/2). (ii) A = kλ(A) for all A ⊂ X. (iii) A′ ∈ S(X) for all A ⊂ X. (iv) For all A ⊂ X, if k(A) ⊂ χ(A), then k(A) = A. The following two results show that the axioms (T1/2) and (T1/4) can also be characterized in terms of properties of ∆. Theorem 5.5. g satisfies (T1/2) iff ∆ = ∆1 ∪ ∆2, where ∆1 ∈ g 2 and ∆2 is saturated. Proof. (⇒) Suppose g satisfies (T1/2). Let A1 = {x ∈ X : {x} ∈ g} and A2 = {x ∈ X : k(x) = {x}}, and let ∆i = ⋃ x∈Ai {x} × {x}, i = 1, 2. By definition of Low separation axioms via the diagonal 49 (T1/2), it is obvious that ∆ = ∆1 ∪ ∆2. Also, ∆1 = ⋃ x∈A1 {x} × {x} ∈ g2 and sat(∆2) = ⋃ x∈A2 k(x) × k(x) = ⋃ x∈A1 {x} × {x} = ∆2. (⇐) Suppose ∆ = ∆1 ∪ ∆2, where ∆1 ∈ g 2 and ∆2 is saturated. Since ∆1 ⊂ ∆, there exists a set B1 ⊂ X such that ∆1 = ⋃ x∈B1 {x} × {x}. Also, ∆1 ∈ g 2 implies that ∆1 = ⋃ α Aα × Bα, with Aα, Bα ∈ g. Thus, for each x ∈ B1, {x} × {x} = Aα × Bα for some α, and visceversa, and it follows that {x} ∈ g for each x ∈ B1. On the other hand, ∆2 = sat(∆2) = ⋃ x∈B2 k(x)×k(x) ⊂ ∆, for some set B2 ⊂ X, and it follows that k(x) = {x}, for each x ∈ B2. It is clear that X = B1 ∪ B2. Therefore g satisfies (T1/2). � Theorem 5.6. g satisfies (T1/4) iff ∆ = ∆1 ∪ ∆2, where ∆1 is kerneled and ∆2 is saturated. Proof. (⇒) Suppose g satisfies (T1/4). Let A1 = {x ∈ X : χ(x) = {x}} and A2 = {x ∈ X : k(x) = {x}} and let ∆i = ⋃ x∈Ai {x} × {x}. It is clear that ∆ = ∆1 ∪ ∆2, χ(∆1) = ∆1, and sat(∆2) = ∆2. (⇐) Suppose ∆ = ∆1 ∪∆2, where ∆1 is kerneled and ∆2 is saturated. There exists B1 ⊂ X such that ∆1 = ⋃ x∈B2 χ(x)×χ(x). Since ∆1 ⊂ ∆, it follows that χ(x) = {x} for each x ∈ B1. Also, there exist B2 ⊂ X such that k(x) = {x}, for each x ∈ B2, and since X = B1 ∪ B2, we conclude that g satisfies (T1/4). � Some results found in [2] and [3] can be obtained directly from those proved here, by considering the generalized topologies given by the α-open sets and the λ-open sets respectively. Acknowledgements. We would like to thank Professor Salvador Garćıa- Ferreira for helpful discussions related to this research. He visited Universidad de Los Andes during the month of April 2006, with a partial support given by the program Intercambio Cient́ıfico. References [1] F. Arenas, J. Dontchev, and M. Ganster, On λ-sets and the dual of generalized conti- nuity, Questions and Answers in Gen. Top.15, no. 1 (1997), 3–13. [2] M. Caldas, D. N. Georgiou, and S. Jafari, Characterizations of low separation axioms via α-open sets and α-closure operator, Bol. Soc. Paran. Mat., 21, no. 1-2(2003), 1–14. [3] M. Caldas and S. Jafari, On some low separation axioms via λ-open sets and λ-closure operator, Rend. Circ. Mat. Palermo, Serie II(Tomo LIV) (2005), 195–208. [4] M. Caldas, S. Jafari and M.M. Kovár, Some properties of θ-open sets, Divulgaciones Matemáticas 12, no. 2(2004), 161–169. [5] A. Csaszar, Separation axioms for generalized topologies, Acta Math. Hungar. 104 (2004), 63–69. [6] N. Levine, Semi-open sets and semicontinuity in topological spaces, Amer. Math. Monthly 70 (1963), 36–41. 50 M. L. Colasante, C. Uzcátegui and J. Vielma [7] H. Maki, Generalized λ-sets and the associated closure operator, The Special Issue in Conmmemoration of Proffesor Kazusada IKEDA’s Retirement, (1986), 139–146. [8] O. Nj̊astad, On some classes of nearly open sets, Pacific. J. Math. 15 (1965), 961–970. [9] N. V. Veličko, H-closed topological spaces, Mat Sb. 70 (1966), 98–112. English transl., Amer. Math. Soc. Transl., 78 (1968), 102–118. Received August 2006 Accepted September 2007 M. L. Colasante (marucola@ula.ve) Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los An- des, Mérida 5101, Venezuela. C. Uzcátegui (uzca@ula.ve) Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los An- des, Mérida 5101, Venezuela. J. Vielma (vielma@ula.ve) Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los An- des, Mérida 5101, Venezuela.