Abstract
We prove that, for an arbitrary Baire space X, a linearly ordered compact set Y, and a separately continuous mapping ƒ: X × Y → R, there exists a G δ-set A ⊆ X dense in X and such that the function ƒ is jointly continuous at every point of the set A × Y, i.e., any linearly ordered compact set is a co-Namioka space.
References
I. Namioka, “Separate continuity and joint continuity,” Pacif. J. Math., 51, No. 2, 515–531 (1974).
G. Debs, “Points de continuite d’une function separement continue,” Proc. Amer. Math. Soc., 97, 167–176 (1986).
A. Bouziad, “Notes sur la propriete de Namioka,” Trans. Amer. Math. Soc., 344, No. 2, 873–883 (1994).
A. Bouziad, “The class of co-Namioka spaces is stable under product,” Proc. Amer. Math. Soc., 124, No. 3, 983–986 (1996).
R. Deville, “Convergence ponctuelle et uniforme sur un espace compact,” Bull. Acad. Pol. Sci., Ser. Math., 37, 507–515 (1989).
J. Calbrix and J. P. Troallic, “Applications separement continues,” C. R. Acad. Sci. A, 288, 647–648 (1979).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 1001–1004, July, 2007.
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Mykhailyuk, V.V. Linearly ordered compact sets and co-Namioka spaces. Ukr Math J 59, 1110–1113 (2007). https://doi.org/10.1007/s11253-007-0071-7
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DOI: https://doi.org/10.1007/s11253-007-0071-7