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HOUSTON JOURNAL OF MATHEMATICS @ 1998 University of Houston Volume 24, No. 4, 1998 SUBMAXIMALITY, EXTREMAL DISCONNECTEDNESS GENERALIZED CLOSED JILING CAO, MAXIMILIAN GANSTER COMMUNICATED ABSTRACT. In this paper, in a topological space. classes of generalized sg-submaximal disconnected we continue the study of generalized we study closed sets coincide. the question closed sets when some A new class of spaces, the class of is also introduced. spaces and sg-submaximal kinds of generalized AND IVAN REILLY BY JUN-IT1 NAGATA In particular, spaces, AND SETS Characterizations of extremally spaces are established via various closed sets. 1. I~vTR~DUCTI~N During the last few years the study erable interest among general of generalized topologists. closed sets has found consid- One reason is these objects are natural generalizations of closed sets. More importantly, generalized closed sets suggest some new separation axioms which have been found to be very useful in the study of certain objects of digital topology, for example, the digital In [5], Dontchev summarized in a diagram the fundamental the various types of generalized closed sets. Concerning posed two questions In the present which have been answered paper, we take another we are able to characterize sets. Moreover, we introduce this class of spaces in terms the notion Key words and phrases. Preclosed, and Reilly in [4]. diagram. In doing so, spaces via generalized of sg-submaximal of the Hewitt 1991 Mathematics Subject Classi&ation. by Cao, Ganster disconnected [S]). relationships between this diagram, he also look at Dontchev’s extremally line (see e.g. closed spaces and investigate decomposition of a topological space. 54A05, 54F65, 54GO5. sg-closed, gcr-closed, sg-submaximal, extremally of the University of Auckland disconnected. The during second author wishes his stay as an Honorary to acknowledge the hospitality Research in the Department Fellow 681 of Mathematics. 682 CAO, This enables us to provide GANSTER an example AND REILLY of an sg-submaximal space which is not g-submaximal. A subset preopen, S of a topological semi-preopen] space (X, 7) is called if S C int(cl(intS)) [resp. a-open [resp. S & cl(intS), semi-open, S C_ int(clS), S c cl(int(clS))]. M oreover, S is said to be o-closed [resp. semi-closed, preclosed, semi-preclosed] if X \ S is o-open [resp. semi-open, preopen, semi-preopen] or, equivalently, if cl(int(clS)) C S [resp. int(clS) S]. The a-closure [resp. semi-closure, o-closed [resp. semi-closed, preclosure, the smallest S. It is well-known S U cl(intS) C S, cl(intS) preclosed, that (Y-clS = S U cl(int(clS)) and spclS = S U int(cl(intS)). C S, int(cl(intS)) semi-preclosure] semi-preclosed] of S C X is set containing and sclS = S U int(clS), Th e o-interior C pclS = of S s X is the largest a-open set contained in S, and we have o-intS = Snint(cl(intS)). It is worth mentioning that the collection of all o-open subsets of (X, r) is a topology rN on X [15] which is finer than semi-open and preopen of dense subsets. Definition. A subset (1) generalized U is open; (2) g-open (3) sg-closed r, and that a subset 1161. M oreover, if and only if it is (X, 7) and (X, T”) share the same class A of (X, r) is called closed (briefly, g-closed) [12], if X \ A is g-closed; [3], if sclA & U whenever [12] if clA C U, whenever A C U and A c U and U is semi-open; [3], if X \ A is sg-closed; (4) sg-open (5) gcx-closed [13], if a-clA C U whenever lently, if A is g-closed in (X, P); (6) gs-closed [2] if sclA s U whenever In [5], a diagram tween S is a-open has been provided A C U and U is a-open, or equiva- A C U and U is open. in order to illustrate the relationships be- these classes of generalized closed sets. Let S be a subset of (X, r). A of S is a pair < El, Ez > of disjoint dense subsets of S. Furthermore, resolution S is said to be resolvable solvable. In addition, S is irresolvable. are codense if it possesses S is called Observe that otherwise Lemma 1.1. [lo, 9] Every space (X,7) S is called irre- if every open subspace if < El, Ez > is a resolution in (X, r), i.e. have empty where F is closed and resolvable a resolution, strongly irresolvable, of S then El and E2 interior. has a unique of decomposition and G is open and hereditarily X = FUG, irresolvable. GENERALIZED Recall that a space CLOSED SETS 683 is said to be submaximal [g-submaximal] (X,7) if ev- Obviously, every submaximal space ery dense subset is open [g-open]. submaximal. Note that every submaximal space is hereditarily irresolvable, that every dense subspace of clG is strongly Lemma 1.1. Any indiscrete not submaximal. In [ll], JankoviC irresolvable, space with at least two points and Reilly pointed out that where is gand G is defined is g-submaximal every singleton in but {z} of a space (X,r) is either nowhere X=XiUXzof(X,r), dense or preopen. This provides another decomposition w h ere Xi = {x E X : {z} is nowhere dense } and X2 = {x E X : {x} is preopen }. Th e usefulness the following result of this decomposition which will be used extensively is illustrated by in the sequel. Theorem 1.2. Let (X, T) be a space, and A be a subset of X. (1) [6] A is sg-closed if and only if X1 n sclA c A. Then (2) pclA 5 Xi U A. PROOF. Then (1) has been {x} is preopen proved in [6]. Now let z E pclA and suppose and thus {x}nA # 0, i.e., z E A. This proves that x 61 Xi. q (2). Throughout this paper, X = F U G and X = Xi U X2 will always denote the Hewitt decomposition of (X, r) and the decomposition due to Jankovic and Reilly, respectively. It is clear that every sg-closed subset of a space (X, r) is gs-closed. However the converse is not true in general. In [14],Maki et al called spaces whose gsclosed sets are sg-closed T&spaces. The following result characterizes the class of T,,-spaces. Theorem 1.3. The following are equivalent for a space (X, r): (1) (X, r) is a TgS-space, (2) every singleton PROOF. {x} of X is either closed or preopen. (1) =s (2). Let x E X1 and suppose is gs-closed, contradiction. (2) + (1). dense and semi-open. Let A be gs-closed x $! A, i.e.A c X \ (z}. Then that {x} is not closed. Then X \ (x} Hence Xi n scl(X \ {x}) = Xi s X \ {x}, a and let x E Xr n sclA. Then sclA C_X \ {CC}, a contradiction. {x} is closed. If q 684 CAO, GANSTER 2. SOME AND REILLY CHARACTERIZATIONS DISCONNECTED Recall that a space Lemma 2.1. (1) (X,7) SPACES (X, T) is eztremally open subset of X is open. due to Njbstad [15]. OF EXTREMALLY disconnected INe first note the following The following is extremally if the closure result of every which is essentially are equivalent for a space (X,r) : disconnected, (2) scl(A U B) = sclA IJ sclB for all A, B C X, (3) the union of two semi-closed As an immediate Theorem 2.2. (1) (X,7) PROOF. The following is extremally (2) the union Lemma consequence (1) + sets is semi-closed. we now have the following theorem. are equivalent for a space (X,7) : disconnected, of two sg-closed sets is sg-closed. (2). Let A and B be sg-closed. Then, by Theorem 1.2 and 2.1, X1 n scl(A U B) = X1 n(sclA = (Xr n SCM) u(Xr AU B is sg-closed, Therefore (2) + (1). by Theorem Let U be open u sclB) n sciB) 2 Au and suppose B. 1.2. there exists 5 E X with x E clU \ int(clU). If Si = U U(x) and Sz = (X \ clU) U{x}, then Si and & are semiopen, hene sg-open. By assumption, Si n S2 = {x} is sg-open, i.e. D = X \ {x} is sg-closed. Clearly x E Xr and so D isdense, i.e. sclD = X. Thus Xi n sclD = 0 Xi c X \ {x}, a contradiction. Referring implications extremally to the diagram in that diagram disconnected in [5] we now consider possible converses of some thereby obtaining some more characterizations of spaces by using some classes of generalized Theorem 2.3. For a space (X, r) the following (1) (X,7) is extremally (2) every semi-preclosed (5’) every sg-closed are equivalent: disconnected, subset of X is preclosed, subset of X is preclosed, (4) every semi-closed subset of X is preclosed, (5) every semi-closed subset of X is a-closed, (6) every semi-closed subset of X is ga-closed. closed sets. GENERALIZED PROOF. Both 685 (3) + (4) and (5) + (6) are obvious. (1) + (2). If S is semi-preclosed, is extremally SETS CLOSED disconnected, then S = spclS = S lJ int(cl(intS)). then cl(intS) = int(cl(intS)). Then Since X S = pclS, i.e. S is preclosed. (2) + preclosed. (3). Prom 2.4 of [5], every Theorem sg-closed subset of X is semi- (4) + (5). This follows directly from the result in [16] that a subset if and only if it is both semi-closed and preclosed. (6) + (1). this end, It suffices to show that let S be regular (6), S is go-closed SUcl(int(clS)). which open. Then implies that It follows that (X, r) is extremally every regular S is both a-clS open To and a-open. C_ S. On the other Therefore, hand By cr-clS = S is closed, and cl disconnected. It has been shown in [4] that (X,F) is g-submaximal preclosed subset is gcu-closed. As a variation of that result Theorem 2.4. For a space (X, r) the following (1) every semi-preclosed (2) (X, P) set S is closed. semi-closed S = cl(int(clS)). is o-closed if and only if every we now obtain are equivalent: set is ga-closed, is extremally disconnected and (X, P) is g-submaximal. PROOF. (1) =$ (2). If every semi-preclosed set is gcr-closed, then every semiclosed set is go-closed and hence by Theorem 2.3, (X,7) is extremally disconnected, Let A C X be rU-dense. so is (X, ra). (X, r), and therefore go-closed. (2) 3 (1). Let A be a semi-preclosed tremally disconnected, then by Theorem Theorem 2.2 in [4] that A is go-closed. 3. Then Thus A is g-open sg-SUBMAXIMAL X \ A is semi-preclosed in in (X, 7”). subset of (X, r). Since (X, r) is ex2.3, A is preclosed. It follows from 0 SPACES Observe that we show in Theorem 2.2 that a space is extremally disconnected if and only if the union of two sg-closed sets is sg-closed. In general, however, the union have of two sg-closed Lemma 3.1. [7] The union sets fails to be sg-closed of a closed subset and an sg-closed Let us call a space (X, r) sg-submaximal is sg-open [o-open]. dense subsets, [3]. On the other hand, We note that, [a-submaximal] we do subset is sg-closed. if every dense subset since (X, 7) and (X, r(2) share their classes of (X, r) is cu-submaximal if and only if (X, ra) is submaximal. This CAO. GANSTER AND REILLY 686 property has been characterized by Ganster [9, Theorem 41. In the following, will characterize sg-submaximality in terms of generalized with a lemma about the Hewitt decomposition. closed sets. Lemma 3.2. Let E C X be a codense subset of a space (X,r), Then int(clE) n clG = 0. PROOF. i.e. we We start intE = 0. Since X \ E is dense in X, (X \ E) n G is dense in G. By Theorem in[9],int(X\E)nG=(X\clE)nG’ 2 IS d ense in G, i.e. G C cl( (X \ clE) n G). Then, int(clE) n G c cl((X \ clE) n G) n int(clE) c cl((X \ clE) n G n int(clE)) and so int(clE) Theorem = 0. 0 n clG = 0. For a space (X, r) the following 3.3. are equivalent: (1) XI c: clG , (2) every preclosed subset is sg-closed, (3) (X, r) is sg-submaximal, (4) (X, rN) is sg-submaximal. PROOF. (1) 3 (2). Let C C X be preclosed. where H is closed and E is codense Xi n sclE = Xi n(E By assumption and Lemma implies that E is sg-closed. 1 in [9], C = H l_l E, By Proposition in (X, r). We have that U int(clE)) = (Xi n E) U(X, n int(clE)). 3.2, Xi n int(clE) = 0. Hence, Xi n sclE C E which By Lemma 3.1, C is sg-closed. (2) 3 (3). Let D be a r-dense subset of X. Then X \ D is preclosed in (X, T), and so is sg-closed. Therefore, D is sg-open and (X,7) is sg-submaximal. (3) @ (4) 1s . ob vlous, since r and ra share the classes of dense subsets classes of sg-open subsets (see [l]). (4) + (1). Let x E Xi and suppose of intF, and without be a resolution codense, it is sg-closed sclEz = E2 U int(clEs) It was pointed Corollary 3.4. If We therefore (X, P) in the o-open G X \ {xc>, a contradiction. out in Theorem only if Xi C: int (clG). that z @ clG, i.e. x E intF. Let < El, Es > loss of generality let x E El. Since Ez is in (X, 7”) and contained = intF and the 2.1 in [4] that (X,ra) set X \ {x}. Hence 0 is g-submaximal have is g-submaximal, then (X, ?) is sg-submaximal. if and GENERALIZED CLOSED SETS Our final example shows that the converse Example 3.5. There is an sg-submaximal 687 of Corollary 3.4 is not true in general. space which is not g-submaximal. It is well-known that there exists a Hausdorff, dense-in-self, submaximal topology 0 on R, the set of reals. Let Y be an infinite set disjoint from R and let X = R (J Y. We now define is a u-open a topology r on X. set containing If x E R \ {0}, a basic neighbourhood x but not 0. has the form U U Y, where If x = 0, a basic U is a a-open set containing of x neighbourhood of x x. If y E Y then y has Y as a minimal open neighbourhood. X,=Y,F=YU{O}andG=R\{O}. Now it is easily checked that X1 = R, M oreover, clG = R and int(clG) = G. Therefore, and so (X, F) X1 C clG but X1 g int(clG), is sg-submaximal but not g-submaximal. Acknowledgement: suggestions The authors wish to acknowledge the useful comments and of the referee(s). REFERENCES [1] F. Arenas, topology, J. Cao, J. Dontchev and M. Puertas, Some covering properties of the CI- preprint. [2] S. Arya and T. Now, Characterizations of s-normal spaces, Indian J. Pure Appl. Math., 21 (1990), 717-719. [3] P. Bhattacharya and B. K. Lahiri, Semi-generalized closed sets in topology, Indian J. Math., 29 (1987), 375-382. [4] J. Cao, M. Ganster and I. L. 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Balachandran, Generalized wciosed sets in topology, Univ. Ed. Part III, 42 (1993), 13-21. 141 H. Maki, K. Balachandran and R. Devi, Remarks on semi-generalized Bull. Fukuoka generalized semi-closed 151 0. Njbstad, closed sets and sets, Kyungpoolc Math. J., 36 (1996), 155-163. On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961-970. 688 CAO, GANSTER [16] I. L. Reilly and M. K. Vamanamurthy, Hungar., 45 AND REILLY On cr-continuity in topological spaces, Acta Math. (1985), 27-32. Received December 25, 1997 (Cao, Ganster and Reilly) DEPARTMENT OF MATHEMATICS,THE UNIVERSITYOF AUCKLAND, PRIVATEBAG ~~O~~,AUCKLAND ~,NEw ZEALAND (Ganster)PERMANENT ADDRESS: DEPARTMENT OF MATHEMATICS, GFLAZ UNIVERSITYOF TECHNOLOGY, STEYRERGASSE 30,A-8010 GRAZ, AUSTRIA