summary:Some results on cleavability theory are presented. We also show some new [16]'s results

Cammaroto, Filippo

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Cleavability and divisibility over developable spaces

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Commentationes Mathematicae Universitatis CarolinaeFilippo CammarotoCleavability and divisibility over developable spacesCommentationes Mathematicae Universitatis Carolinae, Vol. 37 (1996), No. 4, 791--796Persistent URL: http://dml.cz/dmlcz/118885Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1996Institute 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 mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 37,4 (1996)791–796 791Cleavability and divisibility over developable spacesFilippo CammarotoAbstract. Some results on cleavability theory are presented. We also show some new[16]’s results.Keywords: cleavable space, developable space, subdevelopable space, D-regular space,D-completely regular space, D-compact space, D-normal space, divisible spaceClassification: 54A20, 54C10, 54D20, 54E300. Introduction and preliminariesIn 1985 Arhangel’skii in [1], [2], introduced various types of cleavability (orig-inally called splittability) of topological spaces as follows.Let P be a class of topological spaces andM a class of continuous mappings(containing all homeomorphisms). Let A be a subset of a space X . X is saidto be M-cleavable over P along A if there exist a space Y ∈ P and a mappingf ∈ M, f : X → Y , such that Y = f(X) and A = f−1f(A).If A is a family of subsets of X , then we shall say that X isM-cleavable overP along A if it isM-cleavable over P along each A ∈ A. X isM-cleavable overP if it is M-cleavable over P along each A ⊂ X . When P is the family of allsubsets of a given space Y we speak about M-cleavability of X over Y insteadofM-cleavability over P .If X isM-cleavable over P along all singletons {x}, x ∈ X , one speaks aboutpointwise M-cleavability (of X) over P .WhenM is the class of all continuous (open, closed, perfect, . . . ) mappings,we use the term cleavable (open cleavable, closed cleavable, perfectly cleavable,. . . ) over P instead ofM-cleavable over P .In particular, a cleavable space is a space which is cleavable over the class ofall separable metrizable spaces (or equivalently over Rω, because every separablemetrizable space can be embedded into Rω). This case is of particular interest.The paper [7] studied cleavability in details and contains many interested resultsin this connection.The following two questions concerning cleavability are quite natural:This research was supported by a grant form the C.N.R. (G.N.S.A.G.A.) and M.U.R.S.T.through “Fondi 40%”.This paper was presented in the author’s lecture at the I Congreso Iberoamericano de Topolo-gia y sus Applicaciones - Universidad Jaume I, Castellón (Spagna) 28-30 Marzo 1995792 F.CammarotoGeneral question A. Which spaces X are M-cleavable over a class P (alongsubset of X or along a collection of subsets of X)?General question B. If a space X isM-cleavable over P , which properties Xhas? Does X belong to P?Let us denote that if there exists a continuous bijection from X onto a spaceY ∈ P , then, obviously, X is cleavable over P . In this case one can say thatX is absolutely cleavable over P . So, cleavability (over P) may be viewed as ageneralization of continuous bijection (onto some Y ∈ P).A natural question in this connection is: “When cleavability over P implies theexistence of a continuous bijection onto some Y ∈ P?” Here is the lemma (whichis often used for the proofs of many theorems concerning cleavability) about this:Lemma 0.1 ([2]). Let τ be a cardinal, P a class of spaces. Let a space X becleavable over P . If {Aα : α ∈ 2τ} is a collection of pairwise disjoint subsetsof X , then there is a family {Yβ : β ∈ τ} ⊂ P and a continuous mappingf : X →∏{Yβ : β ∈ τ} such that Aα = f−1f(Aα) for each α ∈ 2τ . In particular,if P is hereditary and τ -multiplicative class, then if a space X of cardinality ≤ 2τis cleavable over P , then it is absolutely cleavable over P .One of the most important and useful generalization of metrizable spaces aredevelopable spaces. Recall that a spaceX is developable if there exists a countablecollection {Ui : i ∈ ω} of open covers of X such that for every x ∈ X the family{St(x,Ui) : i ∈ ω} is a local base for X at x. (Here St(x,Ui) is the union of allmembers of Ui containing x.) A spaceX is subdevelopable if it admits a continuousbijection onto a developable T1-space.In 1978, H. Brandenburg began the systematic investigation of topologicalspaces generated by developable spaces (instead of metrizable spaces) and ob-tained some new classes of spaces, as D-completely regular, D-regular, D-compactand so on (for details see Brandenburg’s nice survey [10]). Besides, among devel-opable spaces there is an analogue of the real line, in fact a space, denoted by D1,of cardinality 2ω whose countable power Dω1 is universal for the class Dc of allsecond countable developable T1-space (i.e. every second countable developableT1-space can be embedded into Dω1 ) [10].In this paper we continue the previous two lines of investigation and studycleavability over the class of developable T1-spaces (that generalize metrizablespaces) and over the class of second countable developable T1-space (which gen-eralize separable metrizable spaces); these classes of spaces we shall denote by D,Dc respectively. We clarify which results concerning cleavability over Rω can beor cannot be generalized to the case of cleavability over D and over Dc.Give now some definitions:Definition 0.2 ([9]–[10]). A space X is called:(1) D-regular if each point x ∈ X has a local base consisting of Fσ-sets (notnecessarily open);Cleavability and divisibility over developable spaces 793(2) weakly-D-completely regular if it has a base consisting of open Fσ-sets;(3) D-completely regular if it can be embedded into a product of developableT1-spaces;(4) D-normal (weakly-D-normal) if for every two disjoint closed subsets A andB of X there exists a continuous mapping f from X into some developableT1-space such that f(A) ∩ f(B) = ∅ (f(A) ∩ f(B) = ∅);(5) D-compact if every open cover of X has a finite refinement consisting ofopen Fσ-sets;(6) perfect if every open set is an Fσ-set;(7) R0 when every open set is an union of closed sets.Definition 0.3 ([9]–[10]). A subset A of a topological space X is said to beD-closed iff there exist a continuous mapping (onto) f : X → Y , Y is somedevelopable space, and a closed subset B of Y such that A = f−1(B).Remark 1. Every D-closed subset A of X is a Gδ-set of X .1. Separation axioms and cleavabilityIt is known that if a space X admits a continuous bijection onto a regular (D-regular) space, then X need be regular (D-regular). In this connection we havethe following result.Proposition 1.1. A space X is cleavable over the class P of D-regular (resp. D-completely regular, weaklyD-regular) spaces if and only if X admits a continuousbijection onto some space in P (but X need not be in P).It is known that D-complete regularity is not inversely preserved even underopen perfect mappings and that weak D-complete regularity is not preservedin the preimage direction by perfect mappings. Perfect preimages of D-normalspaces are not necessarily D-normal (see [10]). However we have the followingresult:Proposition 1.2. If a space X is closed pointwise cleavable over the class P ofD-regular (resp. weakly-D-completely regular) spaces, then X ∈ P . If X is closedcleavable over the class of all D-completely regular (D-normal) spaces, then X isalso D-completely regular (D-normal).For a class of spaces the previous result concerning cleavability over the classof weakly D-completely regular may be improved.Theorem 1.1. If a hereditary Lindelöf space X is closed pointwise cleavable overthe class of all weakly D-completely regular space, then X is subdevelopable.794 F.Cammaroto2. Concerning cleavability over D and over DcAs was mentioned, cleavability of a space over the class Dc of second count-able developable T1-space is equivalent to the cleavability of that space over Dω1 .However, this cleavability is equivalent to cleavability over each of the followingtwo classes of spaces:(i) the class of all second countable weakly D-completely regular T1-spaces,(ii) the class of all second countable D-regular T1-spaces.That follows from the fact that these two classes of spaces coincide with theclass Dc.Now we shall give some results regarding cleavability over D1, D and Dc.Proposition 2.1. If a space X is pointwise cleavable over the class D (or overD1), then X is a T1-space of countable pseudocharacter. If X is closed pointwisecleavable over D1, then X is a first countable space.Proposition 2.2. If a space X is perfectly cleavable over D (over Dc or overD1), then X belongs to D (Dc).The following three results are related to General question A.Proposition 2.3. Every space X is cleavable over D (over D1) along each D-closed set (and thus along each D-open set).Since every closed set in a perfect space is D-closed (in fact, a Gδ-set), we have:Proposition 2.4. Every perfect space is cleavable over D along each closed set(and thus, along each open set).Theorem 2.6. Every perfect weakly D-completely regular Lindelöf space X iscleavable over Dc along any disjoint family of open subsets of X .As a nice application of this theorem we have the following result:Corollary 2.7. Let a perfect weakly D-completely regular Lindelöf space Xadmits a perfect mapping onto a space Dc. Then c(X) ≤ ω.Now we give some results devoted to General question B.Proposition 2.8. If a Lindelöf space X is cleavable over Dc, then X is a subde-velopable T1-space (and thus a Gδ-diagonal).Proposition 2.9. A regular Lindelöf space is cleavable over the class Dc if andonly if it is cleavable over the class of separable metrizable spaces.In [7] it was shown that every compact cleavable space is metrizable. Now wegive a generalization of that result.Theorem 2.10. If a H-closed space X is closed cleavable over the class Dc, thenX is subdevelopable.Cleavability and divisibility over developable spaces 795Corollary 2.11. If a minimal Hausdorff space X is closed cleavable over theclass of second countable developable T2-spaces, then X is developable.Recall that a subset A is called D-embedded if every continuous mapping ffrom A into D1 can be extended to a continuous mapping F : X → D1 such thatF |A = f . The following two results should be compared with the correspondingresult in [7] concerning cleavability over Rω.Theorem 2.12. Let X be the union of an increasing sequence X0 ⊂ X1 ⊂ . . . ⊂Xn ⊂ . . . of D-closed of X . If every Xn is cleavable over Dc, then X is alsocleavable over Dc.Theorem 2.13. Let X be a D-completely regular space. If X =⊕{Xα : α ∈2ω} and every Xα is cleavable over Dc, then X is also cleavable over Dc.ONE RESULT ON DIVISIBILITY. We recall the followingDefinition 2.11 ([3]). Let X be a topological space and A be a subset of X . Wesay that a family SA of subsets of X is a divisor (or separator) for A if for everyx ∈ A and every y ∈ X −A there exists S ∈ SA such that x ∈ S and y /∈ S. If allmembers of SA are open (closed) in X , then we say that SA is an open (closed)divisor for A. We say also that a space X is divisible if for every A ⊂ X there isa countable closed divisor for A.Now we shall see one relation between divisibility and cleavability over theclass Dc. Kočinac remarked that a perfectly normal space is divisible if and onlyif it is cleavable (over Rω). Here we have:Theorem 2.12. A perfect space X is divisible if and only if X is cleavableover Dc.3. Some open problemsThe following questions remain open.Question 3.1. Characterize spaces which are cleavable over D1 or over theclass Dc.Question 3.2. If spaces X and Y are cleavable over D or over Dc, is then theproduct X × Y cleavable over the same class?Question 3.3. Characterize D-completely regular spaces X whose D-compacti-fication is cleavable over Dc or over Rω along X .This problem is related to the fact that every D-completely regular space hasa T1-D-compactification (i.e. a D-compact space in which it is dense) [10].796 F.CammarotoReferences[1] Arhangel’skii A.V., A general concept of splittability of topological spaces (in Russian),Abstract Tira. Symp. (1985), Štiinca Kǐsinev, 1985, pp. 8-10.[2] Arhangel’skii A.V., Some new trends in the theory of continuous mapping (in Russian), In:Continuous functions on topological spaces, LGU, Riga, 1986, pp. 5-35.[3] Arhangel’skii A.V., Some problems and lines of investigation in general topology, Comment.Math. Univ. Carolinae 29 (1988), 611–629.[4] Arhangel’skii A.V., A survey on cleavability, to appear.[5] Arhangel’skii A.V., Cammaroto F., On different types of cleavability of topological spaces:pointwise, closed, open and pseudoopen, J. Aust. Math. Soc. (Ser. A) 57 (1995), 1–17.[6] Arhangel’skii A.V., Kočinac Lj., Concerning splittability and perfect mapping, Publ. Inst.Math. (Beograd) 47 (61) (1990), 127–131.[7] Arhangel’skii A.V., Shakhmatov D.B., On pointwise approximation of arbitrary functionsby countable collections of continuous functions, Jour. of Soviet Math. 50.2 (1990), 1497–1511. Translated from Trudy Seminara imeni I.G. Petrovskogo 13 (1988), 206–227.[8] Bella A., Cammaroto F., Kočinac Lj., Remarks on splittability of topological spaces, Q andA in General Topology 9 (1991), 89–99.[9] Brandenburg H., Separation axioms, covering properties and inverse limits generated bydevelopable spaces, Dissertationes Math. 184 (1989), 88.[10] Cammaroto F., On D-completely regular spaces, Supl. Rend. Circ. Mat. Palermo, Ser.II 24(1990), 35–50.[11] Cammaroto F., On splittability of topological spaces, Proc. Brasil Topological Conference,1990.[12] Heldermann N.C.,Developability and some new regularity axioms, Can. J. Math. 33 (1981),641–663.[13] Kočinac Lj., Perfect P-splittability of topological spaces, Zbornik rad. Fil. Fak. (Nis), Ser.Math. 3 (1989), 9–12.[14] Kočinac Lj., Cleavability and divisibility of topological spaces, Atti Acc. Pel. dei Pericolanti(Messina) 70 (1992), 1–16.[15] Kočinac Lj., Cammaroto F., Bella A., Some results on splittability of topological spaces,Atti Acc. Pel. dei Pericolanti (Messina) 68 (1990), 41–60.[16] Kočinac Lj., Cammaroto F., Developable spaces and cleavability, to appear on Rend. Mat.Univ. Roma, 1995.Dipartimento di Matematica, Universita’ di Messina, Contrada Papardo, SalitaSperone, 98168 Messina, Italy(Received October 6, 1995)

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Cleavability and divisibility over developable spaces

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