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One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

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Abstract

We investigate the existence of a separately continuous function f: X × Y → ℝ with a one-point set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x 0X and y 0Y, a separately continuous function f: X × Y → ℝ with the set of discontinuity points {(x 0, y 0)} exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x 0 and y 0, respectively.

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REFERENCES

  1. I. Namioka, “Separate continuity and joint continuity,” Pacif. J. Math., 51, No.2, 515–531 (1974).

    Google Scholar 

  2. Z. Piotrowski, “Separate and joint continuity,” Real Anal. Exch., 11, No.2, 283–322 (1985-1986).

    Google Scholar 

  3. V. K. Maslyuchenko, V. V. Mykhailyuk, and O. V. Sobchuk, “Inverse problems in the theory of separately continuous mappings,” Ukr. Mat. Zh., 44, No.9, 1209–1220 (1992).

    Article  Google Scholar 

  4. V. V. Mykhailyuk, “On the problem of the set of discontinuity points of a separately continuous mapping, ” Mat. Studii, No. 3, 91–94 (1994).

    Google Scholar 

  5. V. K. Maslyuchenko, Relationship between Different Characteristics of the Size of Sets of Points of Joint Continuity of Separately Continuous Mappings [in Ukrainian], Dep. DNTB Ukraine, No. 70-Uk94, Chernivtsi (1994).

  6. O. V. Maslyuchenko, “Oscillations of separately continuous functions on a product of Eberlein compact spaces,” Nauk. Visn. Cherniv. Univ., Ser. Mat., Issue 76, 67–70 (2000).

  7. A. V. Arkhangel’skii, Topological Spaces of Functions [in Russian], Moscow University, Moscow (1989).

    Google Scholar 

  8. V. K. Maslyuchenko, O. V. Maslyuchenko, V. V. Mykhailyuk, and O. V. Sobchuk, “Paracompactness and separately continuous mappings,” in: General Topology in Banach Spaces, Nova Science, New York (2000), pp. 147–169.

    Google Scholar 

  9. V. V. Mykhailyuk, “Dependence of separately continuous functions on \(\mathfrak{n}\) coordinates on products of compact spaces,” Ukr. Mat. Zh., 50, No.6, 822–829 (1998).

    Google Scholar 

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

    Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 94–101, January, 2005.

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Mykhailyuk, V.V. One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces. Ukr Math J 57, 112–120 (2005). https://doi.org/10.1007/s11253-005-0174-y

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  • DOI: https://doi.org/10.1007/s11253-005-0174-y

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