Low separation axioms via the diagonal

@Applied General Topology c Universidad Polit´ecnica de Valencia Volume 9, No. 1, 2008 pp. 39-50 Low separation axioms via the diagonal Mar´ ıa Luisa Colasante, Carlos Uzc´ ategui and Jorge Vielma∗ Abstract. In the context of a generalized topology gon a set X, we give in this article characterizations of some separation axioms between T0and T2in terms of properties of the diagonal in X×X. 2000 AMS Classification: 54A05, 54D10. Keywords: Generalized topologies, intersection structures, envelope operations, kerneled and saturated sets. 1. Introduction A well known elementary fact says that a topological space Xis 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,R0and 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 Xand 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,χgand satg. An example of our results is that gsatisfies T1iff 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. Uzc´ategui and J. Vielma contains the closure of each of its points. It is known that a topology satisfies the axiom R0iff 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 operations. 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 T1/2and T1/4. 2. Preliminaries We follow the notations and definitions used in [5]. A subset gof the power set P(X) of a set Xis a generalized topology (briefly GT ) on Xif {∅, X} ⊆ g and gis closed under arbitrary unions. If gis a generalized topology, then the family of complements of sets in gis usually called an intersection structure. In this article gwill always denote a generalized topology. Definition 2.1 ([5]).An envelope operation on Xis 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) = T{H∈g:A⊆H}. (ii)kg(A) = {x∈X:K∩A6=∅for each K∈gwith x∈K}. (iii)satg(A) = S x∈A kg({x}). The operator χgand kgwere defined in [5] and shown to be envelope operations. It is straightforward to show that satgis an envelope. It is also easy to see that χg(A) = Afor 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 gis a topology, kgis the closure operator cl and χg(A) is the kernel of A, frequently denoted by b Aor Ker(A). Notice that, in general, if τis the topology generated by a GT g, then kg6=clτand χg6=Kerτ(for instance, in Rtake gto 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 arbitrary GT ([5]). (T0) For all x, y ∈X, with x6=ythere is K∈gcontaining precisely one of xand y. Low separation axioms via the diagonal 41 (T1) For all x, y ∈Xwith x6=ythere is K∈gsuch that x∈K, y /∈K. (T2) For all x, y ∈X, x 6=y, there are K, K′∈gsuch 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′∈gdisjoint such that kg(x)⊆Kand kg(y)⊆K′. Proposition 2.3 ([5]). g satisfies (R0)iff for all x, y ∈X, if there is K∈g such that x∈Kand y6∈ K, then there is K′∈gsuch that x6∈ K′and y∈K′. Two of the recently widely studied separation axioms below T1can be stated for a generalized topology gas follows: (T1/2) For all x∈X,{x} ∈ gor {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,kgand satgthat 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 Ais closed (resp. kerneled) iff k(A) = A(resp. χ(A) = A). When gis a topology, kerneled sets are usually called Λ-sets [7]. A subset A⊆Xis said g-saturated (or just saturated) if sat(A) = A, equivalently if k(x)⊆Afor all x∈A. Note that sat(A) is the smallest saturated set containing A, and that Ais saturated if and only if X\Ais kerneled, where X\Adenotes the complement of A. In particular sat(x) = k(x), for any x∈X. The collection of all saturated subsets of Xis denoted by S(X). It is easy to see that S(X) is closed under arbitrary unions and arbitrary intersections. Proposition 2.4 ([5]).Ais closed iff X\A∈g. Proof. Let Aclosed. If y∈X\Athen y /∈k(A),and hence there exists By∈g such that y∈Byand By∩A=∅. Thus X\A=Sy∈X\ABy∈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 g2below is a GT in X2: g2=(D⊆X×X:D=[ α Aα×Bα, with Aα, Bα∈g). In this article the operators k,χand sat on X×Xrefer to the generalized topology g2. Proposition 2.5. For any (x, y)∈X×Xthe 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 ∈gwith x∈Aand y∈B. Then (x, y)∈D=A×B∈g2and so (p, q)∈A×B. Thus p∈χ(x) and q∈χ(y). 42 M. L. Colasante, C. Uzc´ategui and J. Vielma Conversely, if (p, q)∈χ(x)×χ(y) and D∈g2,then (x, y)∈D=SαAα×Bα, with Aα, Bα∈g. There is αsuch that x∈Aαand y∈Bαand hence (p, q)∈Aα×Bα⊆D. This implies that (p, q)∈χ(x, y). Proposition 2.6. (i)If A=Si∈IAi, then χ(A) = Si∈Iχ(Ai). (ii)If A=Si∈IAi, then sat(A) = Si∈Isat(Ai). Proof. (i) was proved in [5] and (ii) is obvious.  Proposition 2.7. Let Abe a subset of X×X. Then (i)χ(A) = S (x,y)∈A χ(x)×χ(y). (ii)sat(A) = S (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∈gsuch that x∈D} kλ(A) = k(A)∩χ(A) kµ(A) = sat(A)∩χ(A) It is easy to see that kθis a weak envelope on Xsuch that k(A)⊆kθ(A) for all A⊆X. Also, it is straightforward to show that kλand kµare envelope operations on X(more generally, the finite intersection of envelope operations is again an envelope). When gis a topology, kθis the well known clθoperator [9, 4] and kλis the clλoperator [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 gis 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)∈Lgiff ∀A∈g[x∈A→y∈A] (x, y)∈Egiff ∀A∈g[x∈A↔y∈A]. Notice that ∆ ⊆Eg⊆Lg. Moreover, Lgis transitive relation and Egis 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)∈Lgiff y∈χ(x)iff x∈k(y). (ii) (x, y)∈Egiff 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∩B6=∅for all A, B ∈gsuch that x∈Aand 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 ∈gwith x∈Aand y∈B. Then (x, y)∈A×B∈g2and thus A×B∩∆6=∅,which implies A∩B6=∅. Reciprocally, let D∈g2containing (x, y). Then D=SαAα×Bα, with Aα, Bα∈g.It follows that (x, y)∈Aα×Bαfor some α. By assumption Aα∩Bα6=∅.If z∈Aα∩Bα,then (z, z)∈D∩∆. Therefore (x, y)∈k(∆). (ii) (x, y)∈χ(∆) = Sx∈Xχ(x)×χ(x) if and only if there is z∈Xsuch that (x, y)∈χ(z)×χ(z) for some z∈Xif 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,χand sat coincide on the relations ∆, Lgand 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)∈Lgand let A, B ∈gwith (x, y)∈A×B. By lemma 3.1, x∈k(y) and thus y∈χ(x). Since χ(x)⊆Athen 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(∆) = Sz∈Xχ(z)×χ(z). Thus, if (x, y)/∈χ(∆), then in particular y /∈χ(x) and this implies that there is A∈gcontaining xsuch that y /∈A, a contradiction. (iii).If (x, y)∈Lgthen, by lemma 3.1(i) and proposition 2.7, (x, y)∈ k(y)×k(y)⊆sat(∆). Hence Lg⊆sat(∆).  44 M. L. Colasante, C. Uzc´ategui and J. Vielma Proposition 3.5. (i)gsatisfies (T2)iff kθ(x) = {x}for each x∈X. (ii)gsatisfies (T1)iff k(x) = {x}for each x∈Xiff χ(x) = {x}for each x∈X. (iii)gsatisfies (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 gsatisfies (T2). Given x∈Xand y6=x, there exist A, B ∈gsuch that x∈A, y ∈Band 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 x6=ythen y /∈kθ(x). Thus (x, y)/∈k(∆) and hence, by lemma 3.2, there exist A, B ∈gsuch that x∈A, y ∈Band A∩B=∅, which shows that gsatisfies (T2). (ii) gsatisfies (T1) iff given x∈Xand y6=x, there exist A∈gsuch that x∈Aand y /∈A, iff given x∈Xand y6=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}=Sy6=xk(y) is saturated for each x∈X, and thus {x}is kerneled for each x∈X. A similar argument shows the reverse implication. (iii) gsatisfies (T0) iff given x∈Xand y6=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)gsatisfies (T2)iff k(∆) = ∆ iff ∆is closed. (ii)gsatisfies (T1)iff χ(∆) = ∆ iff Lg= ∆ iff ∆is saturated iff ∆is kerneled. (iii)gsatisfies (T0)iff Eg= ∆. Proof. (i) By proposition 3.5(i), gsatisfies (T2) iff kθ(x) = {x}for each x∈X, iff y6=ximplies y /∈kθ(x) iff (x, y)/∈k(∆). The second part is obvious. (ii) Suppose gsatisfies (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 x6=y, then (x, y)/∈χ(∆) and thus k(x)∩k(y) = ∅. In particular x /∈k(y), so there exists A∈gsuch that x∈Aand y /∈A. Therefore gsatisfies (T1).The second and third parts follow from lemma 3.1(i) and proposition 3.5(ii). The last part is obvious. (iii) gsatisfies (T0) iff for all x6=y, y /∈χ(x) or x /∈χ(y) iff χ(x)6=χ(y) iff (x, y)/∈Egiff Eg= ∆. Theorem 3.7. g satisfies (R0)iff χ(∆) = Egiff χ(∆) = Lgiff Egis kerneled iff Egis saturated. Proof. Since Eg⊆χ(Eg) = χ(∆),then χ(∆) = Egiff χ(∆) ⊆Eg. From proposition 2.3, gsatisfies (R0) iff x, y ∈Ximplies k(x) = k(y) or k(x)∩k(y) = ∅. Therefore gsatisfies (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 gsatisfies (R0) iff the sets χ(x), x ∈X, form a partition of X, iff sat(∆) = Eg. Theorem 3.8. g satisfies (R1)iff k(∆) = Egiff k(∆) = Lgiff Egis closed. Proof. gsatisfies (R1) iff x, y ∈X, and k(x)6=k(y),implies the existence of A, B ∈gsuch that x∈A,y∈Band A∩B=∅iff (x, y)/∈Egimplies (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)gsatisfies (R0)iff k(x) = χ(x)for each x∈X. (ii)gsatisfies (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 Xis a topological space, and gis 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)⊂Afor every λ-open set A containing x. From this and corollary 3.9, every topological space Xis 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(∆), Lgand 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 Ebe a binary relation on a set X(i.e. E⊆X×X). We will always assume that Econtains the diagonal ∆. We say that a subset A⊆Xis E-saturated if whenever x∈Aand (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)∈Lgiff 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. Uzc´ategui and J. Vielma Lemma 4.2. Let Ebe a binary relation over X. (i)S[E]is closed under arbitrary unions and intersections. (ii)If Eis a symmetric relation, then S[E]is a complete atomic Boolean algebra. Moreover, S[E] = S[F]where Fis the smallest equivalence relation containing Eand 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 ∈Xsuch that y∈Aand (y, x)∈Ebut x /∈A. From the symmetry of E, it follows that (x, y)∈Ewhich implies that x∈A, a contradiction. Let Fbe the transitive closure of E, that is to say, (x, y)∈Fif there are xi∈X,i= 0,··· , n such that x=x0,y=xnand (xi, xi+1)∈E. It is easy to check that Fis 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]Fis E-saturated. Moreover, if z∈A⊆[x]Fand Ais E-saturated, then [z]F= [x]Fand 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 Eis not equal to ∆ and let (x, y)∈Ewith x6=y. Then {y}is not E-saturated.  Remark 4.3. (i)Since k(∆),χ(∆) and Egare 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 gsatisfies T2iff S[k(∆)] = P(X)iff every cofinite set belongs to S[k(∆)]. Clearly the axioms T1and T0are characterized in an analogous way. (ii)If gis 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)gsatisfies (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 gsatisfies (R0) iff Lgis a symmetric relation. Therefore (i)→(ii) follows from lemma 4.2(ii). For the reverse implication, note that if x∈Xand 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∈Xand 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)gsatisfies (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 gsatisfies (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 Aak(∆)-saturated set. Then z∈k(x) and thus (z, x)∈k(∆). By symmetry (x, z)∈k(∆) and as Ais 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 Ais Lg-saturated, then Ais 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. T1/2and T1/4 In this section we characterize the axioms T1/2and T1/4in terms of properties 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 gbe a generalized topology on Xand let ρbe 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∈gsuch that A⊆Hand x /∈H. Let y∈ρ(x)⊂ρ(A). If y∈A, then y∈Hand it must be that x∈Hsince y∈k(x) = ρ(x), a contradiction. Thus y /∈Aand 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)\Aand 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)\Aand A∗=sat(A)\A. The following result is an immediate consequence of lemma 5.1.