We define in this paper the concept of C-space, related withM-spaces and Banach spaces. We obtain various properties on these spaces and propose some open problems

Lupianez, Francisco Gallego

International Mathematical Forum

We define in this paper the concept of C-space, related withM-spaces and Banach spaces. We obtain various properties on these spaces and propose some open problems.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Gallego Lupiáñez, Francisco

Docta Complutense

International Mathematical Forum, 3, 2008, no. 16, 759 - 762On M-Spaces and Banach SpacesFrancisco Gallego LupiañezDept. Mathematics, Univ. Complutense, 28040 Madrid, SpainFG Lupianez@Mat.UCM.EsAbstractWe define in this paper the concept of C-space, related with M-spacesand Banach spaces. We obtain various properties on these spaces andpropose some open problems.Mathematics Subject Classification: 54E18, 54C25, 46B25Keywords: topological spaces, M-spaces, Banach spaces, embedding1 IntroductionThere exist three causes that motive this new paper. First, an early theoremof Corson, also the concept of M-space (defined by K.Morita), and finally apaper on Banach spaces by the author:The Corson Theorem.[3] For any covering U of a infinite dimensionalreflexive Banach space B, where U is formed by bounded, convex sets, there isnot a point x in B such that each neighborhood of x meets only finitely manymembers of U , i.e., U is not locally finite.In our paper [6], we study some problems related to the Corson Theorem.In particular, we proved that:”For every r  0, there exits an open coveringof c0, which is locally finite and is formed by balls of radius r”.We will use in this paper the concept of M-space:Definition 1. [7] A paracompact space X is called a M-space if there issome perfect map from X onto some metric space.760 F. G. Lupiañez2 Main results.Definition 2. Let X be a topological space. We will say that X is a C-spaceif there is some Banach space E and some perfect map f from X onto E suchthat exists a locally finite covering of X formed by pre-images of open balls ofradius 1 by the map f .Remarks. 1. If X is a C-space then X is a paracompact M-space.2. c0, E∞, IRn are C-spaces.Proposition 1. Let X be a topological space, E be a Banach space andf be a perfect map from X onto E. ThenV = {f−1(B1(xj))|j ∈ J} is a locally finite covering of X, if and, only if{B1(xj)|j ∈ J}is a locally finite covering of E.Proof. (⇒) If V covers X also {B1(xj)|j ∈ J} covers E, because f isonto.For each z ∈ E and each x ∈ f−1(z) there exists an open neighborhood Uxzof x, such that meets only finitely members of V. Then {Uxz |x ∈ f−1(z) } isan open covering of f−1(z), and f−1(z) ⊂ ∪rk=1Uxkz (for some x1,...xr ∈ f−1(z))because f is a perfect map.Since f is closed, there exists an open neighborhood W z of z such thatf−1(W z ) ⊂ ∪rk=1Uxkz .Then, f−1(W z ) meets only finitely members of V, and also W z meets onlyfinitely members of{B1(xj)|j ∈ J}.(⇐) If {B1(xj)|j ∈ J} covers E, then V covers X.For each x ∈ X there exists an open neighborhood V f(x) of f(x) such thatmeets only finitely members of{B1(xj)|j ∈ J}.Clearly, f−1(V f(x)) is an openneighborhood of x and meets only finitely members of V.Corollary 1. Let X be a topological space. Then, X is a C-space ifand only if there exists a Banach space E such that has a locally finite opencovering formed by balls of fixed radius, and a perfect map f from X onto E.Corollary 2. For each compact space K, we have that c0×K is a C-space.Proof. Since the projection map p1 is a perfect map from c0 × K ontoc0.Corollary 3. For each compact space K, we have that IRIN × K is aC-space.On M-spaces and Banach spaces 761Proof. It follows from the above Corollary, because c0 is homeomorphicto IRIN ( theorems of Kadec [5] and Anderson [1] ).Proposition 2. Let X be a topological space. If X is separable andC-space, then it is homeomorphic to some closed subset of IRIN × IIN .Proof. Since X is a separable C-space, there is some separable Banachspace E and some perfect map from X onto E. From [8, Theorem 2] it followsthat X is homeomorphic to a closed subset in E × IIN . Finally, therems ofKadec [5] and Anderson [1] yield the conclusion.3 Open problems.1. Let X be a topological space. Have we that X is a C-space if and only ifX is homeomorphic to a closed subset of IRIN × Iw(X)? (where w(X) is theweight of X).2. Have the normed spaces whith locally finite coverings by balls analogousproperties to totally bounded spaces?4 References[1] R.D. Anderson: ”Hilbert space is homeomorphic to the countable infiniteproduct of lines”, Bull. Amer. Math. Soc. 72 (1966), 515-519.[2] C. Bessaga and A. Pelczyn,ski: ”Select topics in infinite-dimensionalTopology”, PWN-Polish Sc.Publ. (Warszawa, 1975).[3] H.H. Corson: ”Collections of convex sets which cover a Banach space”,Fund. Math. 49 (1961), 143-145.[4] R.B. Holmes: ”Geometric fuctional Analysis and its applications”, Spriger-Verlag (New York, 1975).[5] M.I. Kadec: ”On topological equivalence of separable Banach space”,Soviet Math. Dokl. 7 (1966), 319-322.762 F. G. Lupiañez[6] F.G. Lupiáñez: ”Total paracompactness and Banach spaces”, Proc.Amer. Math. Soc. 103 (1988), 210-214.[7] K. Morita: ”Product of normal spaces with metric spaces”, Math. An-nalen 154 (1964), 365-382.[8] J. Nagata: ”A note on M-space and topologically complete space”,Proc. Japan Acad. 45 (1969), 541-543.Received: November 21, 2007

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On M-spaces and Banach spaces

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On M-spaces and Banach spaces

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