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2015 | 3 | 1 |
Tytuł artykułu

The partially pre-ordered set of compactifications of Cp(X, Y)

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R). We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen: (a) Y is not locally compact, (b) X has only one non isolated point and Y is not compact. Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties: (i) X has a non-isolated point with countable character, (ii) X is not pseudocompact, (iii) X is infinite, pseudocompact and Cp(X) is normal, (iv) X is an infinite generalized ordered space. Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2014-09-19
zaakceptowano
2015-04-03
online
2015-07-06
Twórcy
  • Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito exterior
    s/n, Ciudad Universitaria, CP 04510, México D. F., México
  • Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma del Estado de México, Instituto
    Literario No. 100, Col. Centro, C.P. 50000, Toluca, Estado de México, México
  • Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito exterior
    s/n, Ciudad Universitaria, CP 04510, México D. F., México
Bibliografia
  • [1] Arhangelskii A.V., On R-quotient mappings of spaces with a countable base, Soviet Math. Dokl., 1986, 33, 302-305.
  • [2] Chandler R.E., Hausdorff Compactifications, Marcel Dekker, 1976.
  • [3] Contreras-Carreto A., Espacios de funciones continuas del tipo Cp(X, E), PhD thesis, UNAM, 2003.
  • [4] Contreras-Carreto A., Tamariz-Mascarúa Á., On some generalizations of compactness in spaces Cp(X, 2) and Cp(X, Z), Bol.Soc. Mat. Mexicana, 2003, 9(3), 291-308.
  • [5] Dorantes-Aldama A., Tamariz-Mascarúa Á., Some results on weakly pseudocompact spaces, accepted for publication in HoustonJournal of Math.
  • [6] Eckertson R., Sums, products, and mappings of weakly pseudocompact spaces Topology Appl., 1996, 72, 149-157.
  • [7] Engelking R.. General Topology, Sigma Series in Pure Mathematics, 6, Heldermann Verlag Berlin, 1989.
  • [8] García-Ferreira S., García-Máynez A., On weakly-pseudocompact spaces, Houston J. Math., 1994, 20(1), 145-159.
  • [9] Kannan V., Thrivikraman T., Lattices of Hausdorff compactifications of a locally compact space, Pacific J. Math., 1975, 57,441-444.
  • [10] Magill K.D. Jr., The lattice of compactifications of locally compact spaces, Proc. Lond. Math. Soc., 1968, 18(3), 231-234.[Crossref]
  • [11] Porter J., Woods R.G., Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag, 1988.
  • [12] Thrivikraman T., On compactifications of Tychonoff spaces, Yokohama Math. J., 1972, 20, 99-106.
  • [13] Thrivikraman T., On the lattice of compactifications, J. Lond. Math. Soc. (2), 1972, 4, 711-717.[Crossref]
  • [14] Ünlu Y., Lattices of compactifications of Tychonoff spaces, General Topology and its Applications, 1978, 9, 41-57.
  • [15] Visliseni J., Flachsmeyer J.. The power and the structure of the lattice of all compact extensions of a completely regular space,Doklady, 1965, 165(2), 1423-1425.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_taa-2015-0002
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