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Cross Topology and Lebesgue Triples

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The cross topology γ on the product of topological spaces X and Y is the collection of all sets G ⊆ X × Y such that the intersections of G with every vertical line and every horizontal line are open subsets of the vertical and horizontal lines, respectively. For the spaces X and Y from a class of spaces containing all spaces \( {{\mathbb{R}}^n} \), it is shown that there exists a separately continuous function f : X × Y → (X × Y, γ) which is not a pointwise limit of a sequence of continuous functions. We also prove that each separately continuous function is a pointwise limit of a sequence of continuous functions if it is defined on the product of a strongly zero-dimensional metrizable space and a topological space and takes values in an arbitrary topological space.

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  1. 1.

    H. Lebesgue, “Sur l’approximation des fonctions,” Bull. Sci. Math., 22, 278–287 (1898).

  2. 2.

    H. Hahn, Reelle Funktionen. 1. Teil: Punktfunktionen, Acad. Verlagsgesellscheft M.B.H., Leipzig (1932).

  3. 3.

    W. Rudin, “Lebesgue first theorem,” in: L. Nachbin (editor), Mathematical Analysis and Applications, Part B: Advances in Mathematics, Supplementary Studies, Vol. 7b, Academic Press, New York (1981), pp. 741–747.

  4. 4.

    A. K. Kalancha and V. K. Maslyuchenko, “Lebesgue–Cech dimensionality and Baire classification of vector-valued separately continuous mappings,” Ukr. Mat. Zh., 55, No. 11, 1596–1599 (2003); English translation: Ukr. Math. J., 55, No. 11, 1894–1898 (2003).

  5. 5.

    T. Banakh, “(Metrically) quarter-stratifiable spaces and their applications,” Math. Stud., 18, No. 1, 10–28 (2002).

  6. 6.

    O. O. Karlova, “Separately continuous σ-discrete mappings,” Nauk. Visn. Cherniv. Univ., Ser. Mat., Issue 314–315, 77–79 (2006).

  7. 7.

    R. Ellis, “Extending continuous functions on zero-dimensional spaces,” Math. Ann., 186, 114–122 (1970).

  8. 8.

    F. D. Tall, “Stalking the Souslin tree—a topological guide,” Can. Math. Bull., 19, No. 3 (1976).

  9. 9.

    R. Engelking, General Topology [Russian translation], Mir, Moscow (1986).

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Correspondence to O. O. Karlova.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 5, pp. 722–727, May, 2013.

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Karlova, O.O., Mykhailyuk, V.V. Cross Topology and Lebesgue Triples. Ukr Math J 65, 799–805 (2013).

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  • Continuous Function
  • Topological Space
  • Open Covering
  • Hausdorff Space
  • Product Topology