Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Cross Topology and Lebesgue Triples

  • 41 Accesses

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.

This is a preview of subscription content, log in to check access.

References

  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).

Download references

Author information

Correspondence to O. O. Karlova.

Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Karlova, O.O., Mykhailyuk, V.V. Cross Topology and Lebesgue Triples. Ukr Math J 65, 799–805 (2013). https://doi.org/10.1007/s11253-013-0817-3

Download citation

Keywords

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