summary:We provide an answer to a question: under what conditions almost continuity in the sense of Husain implies closure continuity

Alghamdi, M. A.

Saleemi, B. A.

Shahzad, N.

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumB. A. Saleemi; Naseer Shahzad; M. A. AlghamdiAlmost continuity vs closure continuityArchivum Mathematicum, Vol. 37 (2001), No. 1, 39--44Persistent URL: http://dml.cz/dmlcz/107784Terms of use:© Masaryk University, 2001Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 37 (2001), 39 – 44ALMOST CONTINUITY VS CLOSURE CONTINUITYB. A. SALEEMI, N. SHAHZAD AND M. A. ALGHAMDIAbstract. We provide an answer to a question: under what conditions al-most continuity in the sense of Husain implies closure continuity?1. IntroductionAs a generalization of continuity, Levine [7] introduced weak continuity for afunction from one topological space into another. The concept of closure continuity(also called θ-continuity), which implies weak continuity, was given independentlyby Fomin [4] in 1941 and Andrew and Whittlesy [1] in 1966. It appears thatAndrew and Whittlesy were not aware of Fomin’s paper. Other two differentgeneralizations of continuity are the almost continuity of Husain [5] (in Euclideanspaces this notion was initially introduced and studied by Blumberg [2]) and thealmost continuity of Singal and Singal [16]. Similarities and dissimilarities of theseconcepts were discussed in [8]. In 1968 Singal and Singal [16] remarked that closurecontinuous functions are not always almost continuous and asked whether everyalmost continuous function is closure continuous or not. Recently, Saleh [15] gavea positive answer to the above question. Indeed, in 1970s Papp [12] and Kim[6] independently showed that almost continuity implies closure continuity. Itseems that the already published answers ([6], [12]) were not known to Saleh [15].Naturally one may raise the following question:Under what conditions almost continuity in the sense of Husain implies closurecontinuity?In 1974 Noiri [11] considered the above question for weak continuous functionsand showed that any almost continuous function f : X → Y (in the sense ofHusain) is weakly continuous provided it satisfies the following condition:cl(f−1(V ))⊂ f−1 (cl(V )) (1)for each open set V ⊂ Y, where X and Y are topological spaces. Rose [13] observedthat condition (1) is equivalent to weak continuity and so almost continuity inNoiri’s result is redundant. The aim of this paper is to provide an answer to the2000 Mathematics Subject Classification: 54C10.Key words and phrases: almost continuity, closure continuity, weak continuity.Received October 25, 1999.40 B. A. SALEEMI, N. SHAHZAD AND M. A. ALGHAMDIabove question. In fact, we prove that any almost continuous function in the senseof Husain satisfying condition (1) is closure continuous. We observe that everyclosure continuous function satisfies condition (1) but the converse is not true.2. PreliminariesUnless otherwise specified, f : X → Y will denote an arbitrary function of atopological space X into a topological space Y. Let A be a subset of a topologicalspace. Then cl(A) and Int (A) denote the closure and the interior of A respectively.Definition 1. A function f : X → Y is weakly continuous (resp. closure continu-ous) if for each point x ∈ X and each open set V in Y containing f(x), there existsan open set U in X containing x such that f(U ) ⊂ cl(V ) (resp. f (cl(U )) ⊂ cl(V ).The weak continuity of f : X → Y is characterized by condition (1) [13, Theorem7].Remark 1. Every closure continuous function f : X → Y satisfies condition (1).But a function f : X → Y which satisfies condition (1) is not necessarily closurecontinuous, as the following example illustrates:Example 1. Let X = {x, y, z, w} have the topologyτ = {φ, {z}, {z, w}, {x, y, z}, X} .Let Y = {a, b, c, d} have the topologyσ = {φ, {b}, {d}, {a, b},{b, d},{a, b, d},{b, c, d}, Y } .Let f : X → Y be defined byf(x) = a , f(y) = b , f(z) = c , f(w) = d .Thencl(f−1(V ))⊂ f−1 (cl(V ))for all V ∈ σ. But f is not closure continuous (see, for instance, [3]).Definition 2. A mapping f : X → Y is said to be almost continuous in the senseof Singal and Singal if for each point x ∈ X and each open set V in Y containingf(x), there exists an open set U in X containing x such thatf(U ) ⊂ Int cl(V ) .Definition 3. A mapping f : X → Y is said to be almost continuous in thesense of Husain if for each point x ∈ X and each open set V in Y containingf(x), cl(f−1(V ))is a neighbourhood of x.Remark 2.(i) We observe that almost continuity in the sense of Singal and Singal impliesclosure continuity, which in turn implies weak continuity but neither impli-cation is reversible. However, the three notions are equivalent if the rangespace is regular. For details, we refer the reader to [6, 7, 12, 16].(ii) Examples in [9,13,16] show that Definitions 2 and 3 are completely indepen-dent to each other.ALMOST CONTINUITY VS CLOSURE CONTINUITY 41(iii) Example 1 of Rose [13] (or Example 1 of Long and McGehee [9]) shows thatan almost continuous function in the sense of Husain need not be closure con-tinuous. The following example shows that every closure continuous functionis not necessarily almost continuous in the sense of Husain.Example 2. Let X = IR with the countable complement topology. Let Y = {a, b}with the topology {φ, {a}, Y } . Define f : X → Y byf(x) ={a if x is rational,b if x is irrational.Then f is closure continuous but is not almost continuous in the sense of Husain([11, 16]). This shows the independence of closure continuity and almost continuityin the sense of Husain.Definition 4. A function f : X → Y is almost open if f(U ) ⊂ Int cl (f(U )) foreach open subset U of X.Rose [13] obtained that almost openness is equivalent to the following condition:f−1 (cl(V )) ⊂ cl(f−1(V ))for each open subset V of Y.Definition 5. A function f : X → Y is said to be almost open in the sense ofSingal and Singal if the image f(U ) of every regularly open set U in X is open inY.Remark 3. Examples in [13] show the independence of almost openness andalmost openness in the sense of Singal and Singal (see, for instance, [4]).Definition 6. A space X is said to be nearly-compact (resp. closure compact) ifevery open cover of X has a finite sub-family, the interiors of the closures (resp.the closures) of whose members cover X.3. Main ResultsThe following theorem gives a condition which causes almost continuity in thesense of Husain to imply closure continuity for functions.Theorem 1. If f : X → Y is an almost continuous function in the sense ofHusain satisfying condition (1), then f is closure continuous.Proof. Let x ∈ X and let V ⊂ Y be any open set containing f(x). Since fis almost continuous in the sense of Husain, there exists an open set U ⊂ Xcontaining x such thatU ⊂ cl(f−1(V )).Let y ∈ cl(U ). Again, since f is almost continuous in the sense of Husain, for eachopen set W ⊂ Y containing f(y), there exists an open set S ⊂ X containing ysuch thatS ⊂ cl(f−1(W )).42 B. A. SALEEMI, N. SHAHZAD AND M. A. ALGHAMDIAlso, we haveS ∩U 6= ∅because y ∈ cl(U ). It further implies thatS ∩ cl(f−1(V ))6= ∅ .Since S is open, it follows thatS ∩ f−1(V ) 6= ∅ .Furthermore,cl(f−1(W ))∩ f−1(V ) 6= ∅and sof−1 (cl(W )) ∩ f−1(V ) 6= ∅ ,which in turn implies thatcl(W ) ∩ V 6= ∅ .Thus W ∩ V 6= ∅. But W is any open set which contains f(y). Thereforef(y) ∈ cl(V ) .Hencef (cl(U )) ⊂ cl(V )and so f is closure continuous.Remark 4.(i) Theorem 1 shows that almost continuity in the sense of Husain and weakcontinuity together imply closure continuity. It is clear from Example 1(above) and Example 1 of Rose [13] that almost continuity in the sense ofHusain or weak continuity from Theorem 1 cannot be dropped.(ii) If, in addition, Y is regular or Hausdorff and locally compact, then f iscontinuous.Question. Does Theorem 1 hold if condition (1) is replaced by the followingcondition:cl(f−1(V ))⊂ f−1 (cl(V ))for each V ∈ B, where B is an open basis for the topology on Y.Corollary 1. An almost continuous function in the sense of Husain f : X → Yis closure continuous if and only if f satisfies condition (1).Corollary 2. Let f : X → Y be an almost open function in the sense of Singal andSingal satisfying condition (1). If f is almost continuous in the sense of Husain,then f is almost continuous in the sense of Singal and Singal.Proof. If f is almost continuous in the sense of Husain, then by Theorem 1, fis closure continuous. By Theorem 4 of Noiri [10], f is almost continuous in thesense of Singal and Singal.ALMOST CONTINUITY VS CLOSURE CONTINUITY 43Remark 5. For a function which is both almost open (in the sense of Singaland Singal) and almost continuous (in the sense of Husain), the following areequivalent:(i) almost continuity in the sense of Singal and Singal,(ii) closure continuity,(iii) weak continuity,(iv) condition (1).Corollary 3. Let f : X → Y be an almost continuous function in the sense ofHusain satisfying condition (1). If K is a closure compact subset of X, then f(K)is a closure compact subset of Y.Proof. The proof follows from Theorem 1 (above) and Theorem 2 of Saleh [15].Corollary 4. Let f : X → Y be an almost open function in the sense of Singal andSingal satisfying condition (1). If f is almost continuous in the sense of Husainand K ⊂ X is nearly-compact, then f(K) is nearly-compact.Proof. It follows immediately from Theorem 1 (above) and Corollary 3 of Noiri[10].Corollary 5. Let f : X → Y be a function of closure-compact space X onto aUrysohn space X satisfying condition (1). If f is almost continuous in the senseof Husain, then f is almost continuous in the sense of Singal and Singal.Proof. If f is almost continuous in the sense of Husain, then, by Theorem 1, fis closure continuous. By Theorem 5 of Noiri [10], f is almost continuous in thesense of Singal and Singal.References[1] Andrew, D.R. and Whittlesy, E.K., Closure continuity, Amer. Math. Monthly 73 (1966),758–759.[2] Blumberg, H., New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922),113–128.[3] Chew, J. and Tong, J., Some remarks on weak continuity, Amer. Math. Monthly 98 (1991),931–934.[4] Fomin, S.V., Extensions of topological spaces, Dokl. Akad. Nauk SSSR (M.S.) 32 (1941),114–116.[5] Husain, T., Almost continuous mappings, Pr. Mat. 10 (1966), 1–7.[6] Kim, H.O., Notes on C-compact spaces and functionally compact spaces, Kyungpook Math.J. 10 (1970), 75–80.[7] Levine, N., A decomposition of continuity in topological spaces, Amer. Math. Monthly 68(1961), 44–66.[8] Long, P.E. and Carnahan, D.A., Comparing almost continuous functions, Proc. Amer.Math. Soc. 38 (1973), 413–418.[9] Long, P.E. and McGehee, E.E., Properties of almost continuous functions, Proc. Amer.Math. Soc. 24 (1970), 175–180.44 B. A. SALEEMI, N. SHAHZAD AND M. A. ALGHAMDI[10] Noiri, T., Between continuity and weak-continuity, Boll. Un. Mat. Ital. IV. Ser. 9 (1974),647–654.[11] Noiri, T., On weakly continuous mappings, Proc. Amer. Math. Soc. 46 (1974), 120–124.[12] Papp, F.J., Almost- continuous implies θ-continuous, Notices Amer. Math. Soc. 20 (1973),A–175.[13] Rose, D.A., Weak continuity and almost continuity, Internat. J. Math. Math. Sci. 7 (1984),311–318.[14] Rose, D.A., Weak openness and almost openness, Internat. J. Math. Math. Sci. 7 (1984),35–40.[15] Saleh, M., Almost continuity implies closure continuity, Glasgow Math. J. 40 (1998), 263–264.[16] Singal, M.K. and Singal, A.R., Almost continuous mappings, Yokohama Math. J. 16 (1968),63–73.Department of Mathematics, Faculty of ScienceKing Abdul Aziz UniversityP. O. Box 80203, Jeddah-21589SAUDI ARABIAE-mail : naseer shahzad@hotmail.com

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Almost continuity vs closure continuity

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