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2012 | 10 | 2 | 456-465
Tytuł artykułu

Reflecting topological properties in continuous images

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ + for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ + for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight ω 1. Besides, under MA + ¬ CH, the Fréchet-Urysohn property does not reflect in continuous images of weight ω 1 even for compact spaces. An application of our techniques gives a solution of an open problem published by Ramírez-Páramo. If Jensen’s κ +-Axiom $$\left( {\diamondsuit _{\kappa ^ + } } \right)$$ holds for an infinite cardinal κ, then for an arbitrary space X with no G κ-points there exists a continuous surjective map f: X → Y such that w(Y) = κ + and Y has no G tk-points. We apply this result to solve a problem of Kalenda.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
2
Strony
456-465
Opis fizyczny
Daty
wydano
2012-04-01
online
2012-01-18
Twórcy
Bibliografia
  • [1] Alas O.T., Tkachuk V.V., Wilson R.G., Closures of discrete sets often reflect global properties, Topology Proc., 2000, 25(Spring), 27–44
  • [2] Arkhangel’skił A.V., Continuous mappings, factorization theorems and spaces of functions, Trudy Moskov. Mat. Obshch., 1984, 47, 3–21 (in Russian)
  • [3] Arkhangel’skił A.V., Topological Function Spaces, Kluwer, Dordrecht, 1992 http://dx.doi.org/10.1007/978-94-011-2598-7
  • [4] Dow A., An empty class of nonmetric spaces, Proc. Amer. Math. Soc., 1988, 104(3), 999–1001 http://dx.doi.org/10.1090/S0002-9939-1988-0964886-9
  • [5] Engelking R., General Topology, Mathematical Monographs, 60, PWN, Warsaw, 1977
  • [6] Gul’ko S.P., Properties of sets that lie in Σ-products, Dokl. Akad. Nauk SSSR, 1977, 237(3), 505–508 (in Russian)
  • [7] Hajnal A., Juhász I., Having a small weight is determined by the small subspaces, Proc. Amer. Math. Soc., 1980, 79(4), 657–658 http://dx.doi.org/10.1090/S0002-9939-1980-0572322-2
  • [8] Juhász I., Consistency results in topology, In: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, 503–522 http://dx.doi.org/10.1016/S0049-237X(08)71112-1
  • [9] Juhász I., Cardinal Functions in Topology - Ten Years Later, Math. Centre Tracts, 123, Mathematisch Centrum, Amsterdam, 1980
  • [10] Kalenda O.F.K., Note on countable unions of Corson countably compact spaces, Comment. Math. Univ. Carolin., 2004, 45(3), 499–507
  • [11] Ramírez-Páramo A., A reflection theorem for i-weight, Topology Proc., 2004, 28(1), 277–281
  • [12] Tkachenko M.G., Continuous mappings onto spaces of smaller weight, Moscow Univ. Math. Bull., 1980, 35(2), 41–44
  • [13] Tkachuk V.V., Spaces that are projective with respect to classes of mappings, Trans. Moscow Math. Soc., 1988, 139–156
  • [14] Tkachuk V.V., A short proof of a classical result of M.G. Tkachenko, Topology Proc., 2001/02, 26(2), 851–856
  • [15] Tkachuk V.V., A C p-Theory Problem Book, Springer, New York-Dordrecht-Heidelberg-London, 2011 http://dx.doi.org/10.1007/978-1-4419-7442-6
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0141-9
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