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H-CLOSED SPACES AND THE ASSOCIATED 9-CONVERGENCE SPACE T.R. Hamlett* (received 23 May, 1978; revised 25 September, 1978) The concept of 9-convergence was introduced by N. Velicko [8 ], and has also been studied under the name "r-convergence" by L.L. Herrington and P.E. Long [4]. In this paper characterizations of quasi //-closed, C-compact, //-closed, and //-closed Urysohn spaces are obtained in terms of natural conditions on the associated 9-convergence space. Product theorems for quasi //-closed and //-closed spaces are also given. For a given topological space A interior of T-nbd by cl 04) (X, T ) , we denote the closure and and int(4) respectively. (neighbourhood) system at a point 9 -nbd system at U e N(x)} . for every filterbase V e N q (x ) x , denoted A filterbase V e N^(x) F x e X N a (x) , is defined as F e F is said to 9 -accumulate to of all ordered pairs N(x) , and the N a (x) = {cl (i/): is said to 9 -converge to F there exists an there exists an by We denote the F e F x such that { (F,x) : F-^x, x e X} gence space or structure associated with x (f — ► x) , if 0 such that F £. V . A [4], if for every F fl V t <$> . The system is called the 9-conver (X , T ) . The reader is referred to [6 ] for the definitions of 9-cluster point, ^-closure3 and 9 -continuous maps, but we remark that these are merely the closure operator and continuous maps associated with the 9-convergence space. Note that (Z,9(!T)) is a limit space in the most restricted sense of Fischer [3] (principal and pretopological are both terms which are * Author partially supported by a grant from Arkansas Tech. University. Math. Chronicle 8(1979) 83-88. 83 T\ used in the literature to refer to such spaces), and is a gence space in the sense of Ordman [7]. conver We will use [7] as a reference for the general theory of convergence spaces. Definition. Q-closed [ 8 ] if A . A A A subset of a topological space A = clg04) , where is said to be B-open if cl^CA) X - A (X,T) is said to be denotes the Q-closure of Q-closed. is The collection of 0-open sets form the topology associated with the convergence Tq where 9(T) T^ . and is denoted by T is the semi-regularization of topological space (X,T ) [l]. A subset is said to be an H-subset [8 ] of {U for every collection — ^s We remark that : a e A} A 3 of a iX,T) if of open sets such that A c u {U :a 6 A}, there exists a finite subcollection such that — a A c U{cl(t/ ) : i=l,2,...,n} . X is said to be quasi E-closed if it i is an //-subset of itself. A quasi //-closed Hausdorff space is called an H-closed space. topological space Note that these definitions are for an arbitrary (X , T ) with no separation properties assumed. Also note that a subset of a topological space can be an //-subset and not be quasi //-closed as a subspace (see the example on p. 106 of [ 8 ]). If A is a subset of a topological space will denote the elements F A . is a filterbase on general the same as induced on A by (F,x) of Q(T) (X,T ) , then such that x e A Note, however, that 6 (^4 ) » where T' and is not in denotes the subspace topology T . A convergence space is said to be compact if every maximal filterbase converges. Lemma. Let (A,Q(T)a ) (XST) be a topological space with A x . is compact if and only if each filterbase on Then A has a Q-accumulation point. The proof is straightforward and hence omitted. theorem is the main result of this paper. 84 The following A subset Theorem 1. if and only if of a topological space (A3 Q(T)^) Necessity. Proof. A F on A is an H-subset is compact. We suppose that proceed to a contradiction. base (X3T) (i4,0(T)^) is not compact and By the Lemma, there must exist a filter- that does not 9-accumulate to any a e A . Thus for a e A , there exists a U c N(a) and an F e F such that ’ a a F fl cl (U ) = 6 . Now there exists by assumption a finite subcollection a a each {U : i = 1,2, ... ,n} a. of i=l,2,...,n} . ’’ ’ F Now F F o Let ± <i> implies r {U \ a z A] a e F o F o such that F such that fl cl (U a. 4 £ U{cl (£/ ) : a. c f1{F : £=1,2,...,n) . o — a. t ) ? <f> for some 1 5 i 5 n . Hence J n cl(U 1 t <f> , and this is a contradiction. a. a . J J Sufficiency. We show the contrapositive using the Lemma. exists an open cover U = {U a : a e A} A of If there such that A - U{cl(£/ ) : i= l,2,...,n} t for all finite subcollections of ai U , then define F to be the collection of all sets of the form A - U{cl(£/ ) : i=l,2,...,n} F is a filterbase on A that does a. % not 0-accumulate to any point in A , and the proof is complete. Corollary 2. A topological space if is compact. (X3 Q(T)) Corollary 3. (X3 T q ) (XST) If a topological space is quasi H-closed if and only (X3T) is quasi H-closed3 then is compact. The proof follows from corollary 2 and Theorem 3.16 of [7]. Since in general not every closed subset of a quasi //-closed space is quasi //-closed, we have the following definition due to G. Viglino [9]: A topological space is C-compact if each closed subset of the space is an //-subset. 85 Corollary 4. (A3 Q(T)^) A topological space (XST) is C-compact if and only if is compact for every closed subset A of (X>T) . The characterization for ff-closed spaces is given in the following theorem. A convergence space is said to be T2 more than one limit point, and is said to be an element {a;} converges to any point Theorem 5. A topological space Urysohn) space if and only if Proof. Observe that (X,T) (Xt Q(T)) (X,9(T)) is if no filterbase has Tj if no filterbase with y ? x . is an H-closed (H-closed is compact T\{T 2 ) T^(T2) . if and only if (X,T) is Hausdorff (Urysohn), and apply Corollary 2. Theorem 6. The Q-continuous image o f a quasi H-closed space is quasi H-closed. Proof. The proof follows from Corollary 2 and the fact that the continuous image of a compact convergence space is compact. To conclude the paper we use the above theorem to give easy proofs of the following two product theorems. Theorem 7. A nonempty product o f quasi H-closed spaces is quasi H-closed if and only if each factor is quasi H-closed. Proof. Since necessity is an immediate consequence of Theorem 6 , we need only show sufficiency. Let {(Z^,? ) : ot e A) of nonempty quasi #-closed topological spaces. {(^, 0 ( 2 ^ ) ) spaces. : a e A) is a collection of nonempty compact convergence Theorem 3.18 of [7] (Tychonoff's Theorem) then implies that (ttX^ f r O ^ ) ) is compact. For a point form denote a collection Then, by Corollary 2, (a^) e , it is easy to see that all sets of the tt{cl (Z/ ) : £=l, 2 ,...,n} * v{Xn : 3 ^ a.} a. 3 t 86 constitute the nbd system at (x^) with respect to both Consequently, we have that space. Therefore (ttZ ^ ttT^) ©(irT^) (ttX^, ©(ttT^) ) and ttG ^) . is a compact convergence is quasi //-closed by Corollary 2, and the proof is complete. Theorem 8. A nonempty product of H-closed spaces is H-closed if and only if each factor is H-closed. Proof. The proof follows readily from the previous theorem and Theorem 1.3 of [2]. REFERENCES 1. M.P. Berri, J.R. Porter, and R.M. Stephenson, Jr., A survey o f minimal topological spaces3 General Topology and its Relations to Modern Analysis and Algebra, 111 (Proc. Conf. Kanpur, 1968), Academia, Prague, 1971, 93-114. 2. J. Dugundji, Topology} Allyn and Bacon, Boston 1966. 3. H.R. Fischer, Limesraume, Math. Ann. 137(1959), 269-303. 4. L.L. Herrington and Paul E. Long, Characterizations o f C-compact spaces j 5. Proc. Amer. Math. Soc. 52(1975), 417-426. Paul E. Long, An Introduction to General Topology} Charles E. Merrill Publishing Co., Columbus, 1971. 6. Takashi Noiri, Properties o f 0 -continuous functions, Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(1975), No. 6, 887-891. 7. Edward T. Ordman, Convergence and Abstract spaces in Functional Analysis (Part 1)3 Journal of Undergraduate Mathematics, Sept., (1969), 79-95. 87 8. N. Velicko, H-closed. topological spaces3 Amer. Math. Soc. Tra n s l . 78(2) (1969), 103-118. 9. G. Viglino, C-compact spaces3 Duke Math. J. 36(1969), 761-764. 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