On functionally θ-normal spaces

@Applied General Topology c Universidad Polit´ecnica de Valencia Volume 6, No. 1, 2005 pp. 1-14 On functionally θ-normal spaces J. K. Kohli and A. K. Das Abstract. Characterizations of functionally θ-normal spaces including the one that of Urysohn’s type lemma, are obtained. Interrelations among (functionally) θ-normal spaces and certain generalizations of normal spaces are discussed. It is shown that every almost regular (or mildly normal ≡κ-normal) θ-normal space is functionally θ-normal. Moreover, it is shown that every almost regular weakly θ-normal space is mildly normal. A factorization of functionally θ-normal space is given. A Tietze’s type theorem for weakly functionally θ-normal space is obtained. A variety of situations in mathematical literature wherein the spaces encountered are (functionally) θ-normal but not normal are illustrated. 2000 AMS Classification: Primary: 54D10, 54D15, 54D20; Secondary: 14A10. Keywords: θ-closed (open) set, regularly closed (open) set, zero set, regular Gδ-set, (weakly) (functionally) θ-normal space, (weakly) θ-regular space, almost regular space, mildly normal (≡κ-normal) space, almost normal space, δ-normal space, δ-normally separated space, Zariski topology, (distributive) lattice, complete lattice, affine algebraic variety, projective variety. 1. INTRODUCTION Normality is an important topological property and hence it is of significance both from intrinsic interest as well as from applications view point to obtain factorizations of normality in terms of weaker topological properties. First step in this direction was taken by Viglino [18] who defined seminormal spaces, followed by the work of Singal and Arya [13] who introduced the class of almost normal spaces and proved that a space is normal if and only if it is both a seminormal space and an almost normal space. A search for another decomposition of normality motivated us to introduce in [6] the class of (weakly) θ-normal spaces and (weakly) functionally θ-normal spaces. These weak forms of normality serve as a necessary ingredient towards a decomposition of normality. 2 J. K. Kohli and A. K. Das In [6] functionally θ-normal spaces are defined in terms of the existence of certain continuous real-valued functions. In this paper, in analogy with the normal spaces, we obtain a characterization of functionally θ-normal spaces in terms of separation of certain closed sets by θ-open sets. The resulting characterizations are then used to investigate the interrelations that exist among certain generalizations of normal spaces such as (weakly) (functionally) θ-normal spaces, δ-normal spaces defined by Mack [10], δ-normally separated spaces initiated by Mack [10] and Zenor [19], mildly normal spaces (≡κ-normal spaces) studied by Stchepin [15], and Singal and Singal [14], and almost normal spaces studied by Singal and Arya [13]. Moreover, we obtain a decomposition of functional θ-normality in terms of weak functional θ-normality. In the process we obtain improvements of several known results in the literature. Furthermore, a Tietze’s type theorem for weakly functionally θ-normal spaces is obtained. Section 2 is devoted to basic definitions and preliminaries. Characterizations of functionally θ-normal spaces are obtained in Section 3. A Tietze’s type extension theorem is included in Section 4. Interrelations among (weakly) (functionally) θ-normal spaces and certain other weak variants of normality are investigated in Section 5. Moreover, a factorization of functional θ-normality in terms of weak functional θ-normality is obtained in Section 5. Finally, an appendix (Section 6) is included which exhibits a wide variety of situations encountered in mathematical literature wherein a space may be (functionally) θ-normal but not normal. Throughout the paper, no separation axioms are assumed unless explicitly stated otherwise. The closure of a set Awill be denoted by Aand interior by intA. 2. BASIC DEFINITIONS AND PRELIMINARIES Definition 2.1 ([17]).Let Xbe a topological space and let A⊂X. A point x∈ Xis called a θ-limit point of Aif every closed neighbourhood of xintersects A. Let Aθdenote the set of all θ-limit points of A. The set Ais called θ-closed if A=Aθ. The complement of a θ-closed set is referred to as a θ-open set. Lemma 2.2 ([17]).An arbitrary intersection of θ-closed sets is θ-closed and the union of finitely many θ-closed sets is θ-closed. In general the θ-closure operator is not a Kuratowski closure operator, since θ-closure of a set may not be θ-closed (see [4]). However, the following modification yields a Kuratowski closure operator. Definition 2.3. Let Xbe a topological space and let A⊂X. A point x∈Xis called a uθ-limit point of Aif every θ-open set Ucontaining xintersects A. Let Auθ denote the set of all uθ-limit points of A. Lemma 2.4. The correspondence A→Auθ is a Kuratowski closure operator. It turns out that the set Auθ is the smallest θ-closed set containing A. On functionally θ-normal spaces 3 Definition 2.5 ([2]).A function f:X→Yis said to be θ-continuous if for each x∈Xand each open set Ucontaining f(x), there exists an open set V containing xsuch that f(V)⊂U. The concept of θ-continuity is a slight generalization of continuity and is useful in studying spaces which are not regular. Lemma 2.6. Let f:X→Ybe a θ-continuous function and let Ube a θ-open set in Y. Then f−1(U)is θ-open in X. Lemma 2.7 ([6, 7]).A subset A of a topological space X is θ-open if and only if for each x∈Athere exists an open set Usuch that x∈U⊂U⊂A. Definition 2.8. A subset Aof a topological space Xis called regular Gδ-set if Ais the intersection of countably many closed sets whose interiors contain A. The following lemma seems to be known and is easily verified. Lemma 2.9. In a topological space, every zero set is a regular Gδ-set and every regular Gδ-set is θ-closed. Definition 2.10 ([13]).A topological space Xis said to be almost normal if every pair of disjoint closed sets one of which is regularly closed are contained in disjoint open sets. Definition 2.11 ([14, 15]).A topological space Xis said to be mildly normal (≡κ-normal) if every pair of disjoint regularly closed sets are contained in disjoint open sets. 3. FUNCTIONALLY θ-NORMAL SPACES Definition 3.1 ([6]).A topological space Xis said to be (i) θ-normal if every pair of disjoint closed sets one of which is θ-closed are contained in disjoint open sets; (ii) Weakly θ-normal if every pair of disjoint θ-closed sets are contained in disjoint open sets; (iii) Functionally θ-normal if for every pair of disjoint closed sets A and Bone of which is θ-closed there exists a continuous function f: X→[0,1] such that f(A)= 0 and f(B)=1; (iv) Weakly functionally θ-normal (wf θ-normal) if for every pair of disjoint θ-closed sets Aand Bthere exists a continuous function f:X→[0,1] such that f(A)= 0 and f(B)= 1; and (v) θ-regular if for each closed set Fand each open set Ucontaining F, there exists a θ-open set Vsuch that F⊂V⊂U. Definition 3.2. A topological space Xis said to be (i) δ-normal [10] if every pair of disjoint closed sets one of which is a regular Gδ-set are contained in disjoint open sets; and 4 J. K. Kohli and A. K. Das (ii) δ-normally separated [10, 19] if for every pair of disjoint closed sets Aand Bone of which is a zero set there exists a continuous function f:X→[0,1] such that f(A)=0 and f(B)= 1. The following implications are immediate in view of definitions and Lemma 2.9 and well illustrate the interrelations that exist among generalizations of normality outlined in Definitions 3.1 and 3.2. None of the above implications is reversible as is shown by Examples 3.6, 3.7, 3.8 in [6], Examples [10, p. 267, p.270] and Example 3.4 in the sequel. It is shown in [6] that in the class of Hausdorff spaces, the notions of θnormality and functional θ-normality coincide with normality and that in the class of θ-regular spaces all the four variants of θ-normality characterize normality. Furthermore, it is shown in [6] that every Lindel¨of space as well as every almost compact space is weakly θ-normal. In contrast the class of functionally θ-normal space is much larger than the class of normal spaces (see Section 6). Our next result shows that a Urysohn type lemma holds for functionally θ-normal spaces. Theorem 3.3. For a topological space X, the following statements are equivalent. (a) Xis functionally θ-normal. (b) For every pair of disjoint closed sets one of which is θ-closed are contained in disjoint θ-open sets. (c) For every θ-closed set Aand every open set Ucontaining Athere exists aθ-open set V such that A⊂V⊂Vuθ ⊂U. (d) For every closed set Aand every θ-open set Ucontaining Athere exists aθ-open set Vsuch that A⊂V⊂Vuθ ⊂U. (e) For every pair of disjoint closed sets Aand B, one of which is θclosed there exist θ-open sets Uand Vsuch that A⊂U,B⊂Vand Uuθ ∩Vuθ =φ. On functionally θ-normal spaces 5 Proof. To prove the assertion (a) ⇒(b), let Xbe a functionally θ-normal space and let A,Bbe disjoint closed sets in X, where Bis θ-closed. By functional θ-normality of Xthere exists a continuous function f:X→[0,1] such that f(A) = 0 and f(B) = 1. Since every continuous function is θ-continuous, by Lemma 2.6, f−1[0, 1/2) and f−1(1/2, 1] are disjoint θ-open sets containing A and Brespectively. To Prove (b) ⇒(c), let Abe a θ-closed set in Xand let Ube an open set containing A. Since Aand X−Uare disjoint, by hypothesis there exist disjoint θ-open sets Vand Wsuch that A⊂Vand X−U⊂W. So A⊂V⊂X−W⊂ U. Since X−Wis θ-closed and Vuθ is the smallest θ-closed set containing V, A⊂V⊂Vuθ ⊂U. To prove (c) ⇒(d), let Abe a closed set contained in a θ-open set U. Then X−Uis a θ-closed set contained in the open set X−A. By hypothesis, there exists a θ-open set Wsuch that X−U⊂W⊂Wuθ ⊂X−A. Let V=X−Wuθ. Then A⊂V⊂X−W⊂U. Since X−Wis θ-closed and Vuθ is the smallest θ-closed set containing V,A⊂V⊂Vuθ ⊂U. To prove (d) ⇒(e), let Abe a closed set disjoint from a θ-closed set B. Then X−Bis a θ-open set containing A. So there exists a θ-open set Wsuch that A⊂W⊂Wuθ ⊂X−B. Again by hypothesis there exists a θ-open set Usuch that A⊂U⊂Uuθ ⊂W⊂Wuθ ⊂X−B. Let V=X−Wuθ, then U and Vare θ-open sets containing Aand Brespectively and Uuθ ∩Vuθ =∅. The assertion (e) ⇒(c) is easily verified. Finally to prove the implication (c) ⇒(a), let Abe a θ-closed set disjoint from a closed set B. Then A⊂X−B=U1(say). Since U1is open, there exists a θ-open set U1/2such that A⊂U1/2⊂(U1/2)uθ ⊂U1. Again, since (U1/2)uθ is a θ-closed set, there exist θ-open sets U1/4and U3/4such that A⊂ U1/4⊂(U1/4)uθ ⊂U1/2and (U1/2)uθ ⊂U3/4⊂(U3/4)uθ ⊂U1. Continuing the above process, we obtain for each dyadic rational r, a θ-open set Ursatisfying r < s implies (Ur)uθ ⊂Us. Let us define a mapping f:X→[0,1] by f(x) = inf {x:x∈Ur}if xbelongs to some Ur, 1 if xdoes not belongs to any Ur. Clearly fis well defined and f(A) = 0, f(B) = 1. Now it remains to prove that fis continuous. To this end we first observe that if x∈Ur, then f(x)≤r. Similarly f(x)≥rif x /∈(Ur)uθ. To prove continuity, let x∈Xand (a,b) be an open interval containing f(x). Now choose two dyadic rationals pand q such that a < p < f(x)< q < b. Let U=Uq−(Up)uθ. Then Uis an open set containing x. Now for y∈U,y∈Uq. So f(y)≤q. Also as y∈U,y /∈(Up)uθ. Thus f(y)≥q. And so f(y)∈[p, q]. Therefore f(U)⊂[p, q]⊂(a, b). Hence f is continuous.  Example 3.4. A θ-normal space which is not functionally θ-normal. Let Xbe the set of positive integers. Define a topology Ton X, where every odd integer is open and a set Uis open if for every even integer p∈U, the successor and the predecessor of palso belongs to U. Let Y=X∪{∞} be the one point compactification of the space X. Since every paracompact space is θ-normal 6 J. K. Kohli and A. K. Das [6, Theorem 3.12], Yis θ-normal. The space Yis not functionally θ-normal since the θ-closed set {∞} and the closed set {1,2}cannot be separated by disjoint θ-open sets in Y. The above example shows that even a compact θ-normal space need not be functionally θ-normal, and so fills a gap left in [6]. 4. A TIETZE TYPE THEOREM In this section we formulate a Tietze’s type extension theorem for weakly functionally θ-normal spaces. First we introduce the notion of a θ-embedded subset which is instrumental in the formulation of Tietze type theorem. Definition 4.1. A subset A of a topological space Xis said to be θ-embedded in Xif every θ-closed set in the subspace topology of Ais the intersection of A with a θ-closed set in X; equivalently A⊂Xis θ-embedded in Xif every θ-open set in the subspace topology of Ais the intersection of Awith a θ-open set in X. Remark 4.2. Aθ-closed subset of a topological space Xneed not be θembedded in X. For example, let us consider the closed unit interval X= [0,1] with Smirnov’s deleted sequence topology [16, p. 86]. Let A=K∪{0}, where K={1/n:n∈N}. Here Ais a θ-closed subset of Xwhich is not θ-embedded in X. Theorem 4.3. Let Xbe a weakly functionally θ-normal space and let Abe a θ-closed, θ-embedded subset of X. Then (a) every continuous function f:A→[0,1] defined on the set Acan be extended to a continuous function g:X→[0,1]. (b) every continuous function f:A→Rcan be extended to a continuous function g:X→R. Proof. We shall prove the theorem only in case (a). Let Xbe a weakly functionally θ-normal space and let Abe a θ-closed set in X. Let A1={x∈A: f(x)≥1/3}and B1={x∈A:f(x)≤ −1/3}. Then A1and B1are two disjoint θ-closed sets in A. Since Ais θ-embedded in X,A1=F1∩Aand B1=F2∩A, where F1and F2are θ-closed sets in X. Since Ais θ-closed in X, by Lemma 2.2, A1and B1are disjoint θ-closed sets in X. By weak functional θ-normality of X, there exists a continuous function f1:X→[−1/3,1/3] such that f1(A1) = 1/3 and f1(B1) = −1/3. Now, for each x∈A, it is clear that |f(x)−f1(x)| ≤ 2/3. So f−f1is a mapping of Ainto [-2/3, 2/3]. Let g1=f−f1, Then A2={x∈A:g1(x)≥2/9}and B2={x∈A: g1(x)≤ −2/9}are two disjoint θ-closed sets in A. As argued earlier, A2and B2are θ-closed sets in X. Again, by weak functional θ-normality of Xthere exists a continuous function f2:X→[−2/9,2/9] such that f2(A2) = 2/9 and f2(B2) = −2/9. Clearly, |(f−f1)−f2| ≤ (2/3)2on A. Continuing this process, we obtain a sequence of continuous functions {fn} defined on Asuch that |f− n P i=1 fi| ≤ (2/3)non A. It is routine to verify that On functionally θ-normal spaces 7 the function g:X→Rdefined by g(x) = ∞ P i=1 fi(x) for every x∈X, is the desired continuous extension of f. Corollary 4.4. Let Xbe a functionally θ-normal space. Then every continuous function f:A→[0,1] (f:A→R) defined on a θ-closed, θ-embedded subset A of X can be extended to a continuous function g:X→[0,1] (g:X→R). 5. INTERRELATIONS In this section we exhibit the relationships that exists among (functionally) θ-normal spaces, mildly normal spaces and δ-normally separated spaces etc. in the presence of additional mild conditions. This in turn yields improvements of certain known results in the literature. Definition 5.1 ([12]).A topological space Xis said to be almost regular if every regularly closed set and a point out side it are contained in disjoint open sets. The following characterization of almost regular spaces obtained in [8] will be useful in the sequel. Theorem 5.2 ([8]).A topological space Xis almost regular if and only if, for every open set Uin X, int Uis θ-open. Theorem 5.3 ([6]).A topological space Xis θ-normal if and only if for every pair of disjoint closed sets A,Bone of which is θ-closed, there exist disjoint open sets Uand Vsuch that A⊂U,B⊂Vand U∩V=∅. Theorem 5.4. An almost regular, θ-normal space is functionally θ-normal. Proof. Let Xbe an almost regular, θ-normal space. Let Abe a θ-closed set disjoint from a closed set B. Since Xis θ-normal, by Theorem 5.3, there exist disjoint open sets Uand Vsuch that A⊂U,B⊂Vand U∩V=∅. Now A⊂U⊂intU⊂Uand B⊂V⊂intV⊂V. Since Xis almost regular, by Theorem 5.2, intUand intVare disjoint θ-open sets containing A and Brespectively. So in view of Theorem 3.3, the space Xis functionally θ-normal.  Corollary 5.5. An almost regular, θ-normal space is δ-normally separated. Proof. Every functionally θ-normal space is δ-normally separated.  Corollary 5.6. An almost regular paracompact space is functionally θ-normal. Proof. It is shown in [6] that, every paracompact space is θ-normal. Hence the result follows by Theorem 5.4.  Theorem 5.7. A mildly normal, θ-normal space is functionally θ-normal. 8 J. K. Kohli and A. K. Das Proof. Suppose Xis a mildly normal, θ-normal space and let Aand Bbe disjoint closed sets in X, where Ais θ-closed . Since Xis θ-normal, by Theorem 5.3, there exists disjoint open sets Uand Vsuch that A⊂U,B⊂V and U∩V=∅. Then Uand Vare disjoint regularly closed sets in X. Since Xis mildly normal, by [13, Theorem 3], there exists a continuous function f:X→[0,1] such that f(U) = 0 and f(V) = 1. Consequently f(A) = 0 and f(B) = 1 and so Xis a functionally θ-normal space.  Corollary 5.8. A mildly normal paracompact space is functionally θ-normal. Proof. Every paracompact space is θ-normal [6].  Corollary 5.9. A mildly normal, θ-normal space is δ-normally separated. Proof. Every functionally θ-normal space is δ-normally separated.  Remark 5.10. The example of Smirnov’s deleted sequence topology [16, p.86] shows that the hypothesis of θ-normal space in Theorem 5.7 can not be weakened to even “weakly functionally θ-normal space”. Remark 5.11. Example 3.4 shows that the hypothesis of almost regularity in Theorem 5.4 can not be omitted. The same example also shows that the hypothesis of mild normality in Theorem 5.7 can not be deleted. Moreover, simple examples can be given to show that even an almost regular compact space need not be normal. Theorem 5.12. An almost regular weakly θ-normal space is mildly normal. Proof. Suppose Xis an almost regular weakly θ-normal space and let Aand Bbe disjoint regularly closed sets in X. Since Ais regularly closed, X−Ais regularly open and so X−A= int(X−A). By Theorem 5.2, X−Ais θ-open and hence Ais θ-closed. Similarly, Bis θ-closed. Since Xis weakly θ-normal, there exist disjoint open sets Uand Vcontaining Aand B, respectively.  Corollary 5.13 ([14]).An almost regular almost compact space is mildly normal. Proof. It is observed in [6] that every almost compact space is weakly θ-normal.  Corollary 5.14 ([14]).An almost regular Lindel¨of space is mildly normal. Proof. Every Lindel¨of space is weakly θ-normal [6].  Corollary 5.15 ([11]).An almost compact Urysohn space is mildly normal. Proof. Every almost compact Urysohn space is almost regular [11].  Theorem 5.16. An almost regular θ-normal space is almost normal. On functionally θ-normal spaces 9 Proof. Suppose Xis an almost regular, θ-normal space. Let Aand Bbe disjoint closed sets in Xsuch that Ais regularly closed. In view of Theorem 5.2 (as also shown in the proof of Theorem 5.12), Ais θ-closed. So by θ-normality, there exist disjoint open sets Uand Vcontaining Aand B, respectively. Thus Xis almost normal.  Corollary 5.17 ([13]).An almost regular paracompact space is almost normal. Proof. Every paracompact space is θ-normal [6].  Theorem 5.18. An almost regular weakly θ-normal space is weakly functionally θ-normal. Proof. Let Xbe an almost regular weakly θ-normal space and let Aand B be disjoint θ-closed sets in X. Then there exist disjoint open sets Uand V containing Aand B, respectively. It is easily verified that the sets intUand intVare disjoint. By Theorem 5.2, each of the sets intUand intVis θ-open. Thus every pair of disjoint θ-closed sets in Xare separated by disjoint θ-open sets and so by [7, Theorem 8], Xis weakly functionally θ-normal.  Corollary 5.19. An almost regular almost compact space is weakly functionally θ-normal. Corollary 5.20. An almost regular Lindel¨of space is weakly functionally θnormal. Next we give a factorization of functionally θ-normal space in terms of weakly functionally θ-normal space. Definition 5.21. A topological space Xis said to be weakly θ-regular if for each θ-closed set Fand each open set Ucontaining F, there exists a θ-open set Vsuch that F⊂V⊂U. Every θ-regular space is weakly θ-regular. However, the cofinite topology on an infinite set yields a weakly θ-regular space (vacuously) which is not θ-regular. Theorem 5.22. A topological space X is functionally θ-normal if and only if it is both a weakly θ-regular space and a wf θ-normal space. Proof. Necessity is immediate in view of Theorem 3.3 and the fact that every functionally θ-normal space is wf θ-normal. To prove sufficiency suppose that Xis a weakly θ-regular space and a wf θ-normal space. Let Aand Bbe disjoint closed sets in Xsuch that Ais θ-closed. Then A⊂X−Band X−B is open. So by weakly θ-regularity of X, there exists a θ-open set Usuch that A⊂U⊂X−B. Then Aand X−Uare disjoint θ-closed sets and B⊂X−U. Since Xis wf θ-normal, there exists a continuous function f:X→[0,1] such that f(A) =0 and f(X−U) = 1. Since B⊂X−U,Xis functionally θ-normal.  Theorem 5.23. A topological space X which is both a weakly θ-regular space and a weakly θ-normal space is θ-normal.