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Quasicontinuous and separately continuous functions with values in Maslyuchenko spaces. (English) Zbl 1376.54016

The paper generalizes some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions in certain generalized metric spaces. A function \(f:X\times Y\) is called a \(KC\)-function (\(QC\)-function) if it is quasicontinuous (lower quasicontinuous) with respect to the first variable and continuous with respect to the second variable. A topological space \(Z\) is called \(KC\)-Maslyuchenko (\(QC\)-Maslyuchenko) if for each \(KC\)-function (\(QC\)-function) \(f:X\times Y\to Z\) defined on the product of a topological space \(X\) and a second-countable space \(Y\) the set \(D(f)\) of discontinuity points of \(f\) has meager projection on \(X\). A topological space \(Z\) is called \(CC\)-Maslyuchenko if for each separately continuous function \(f:X\times Y\to Z\) defined on the product of separable metrizable spaces \(X\) and \(Y\) the set \(D(f)\) has meager projection on \(X\).
In the paper, the classes of \(KC\)-Maslyuchenko, \(QC\)-Maslyuchenko and \(CC\)-Maslyuchenko spaces are studied. Among many other things, it is shown that the class of \(QC\)-Maslyuchenko spaces is closed under taking countable Tychonoff products (under taking fibered preimages, under strong \(\bar{\sigma}\)-sums, under \(\bar{G}_\delta\)-retral unions, under taking Michael modifications, under taking sequentially admissible function spaces).

MSC:

54C08 Weak and generalized continuity
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54E20 Stratifiable spaces, cosmic spaces, etc.
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[1] Baire, R., Sur les fonctions de variables reélles, Ann. Math. Pures Appl., 3, 1-123 (1899) · JFM 30.0359.01
[2] Banakh, T.; Bogachev, V.; Kolesnikov, A., \(k^\ast \)-metrizable spaces and their applications, J. Math. Sci. (N.Y.), 155, 4, 475-522 (2008) · Zbl 1332.54180
[3] Banakh, T., Quasicontinuous functions with values in Piotrowski spaces, Real Analysis Exchange, 43, 1 (2018) · Zbl 1400.54030
[4] Blass, A., Combinatorial cardinal characteristics of the continuum, (Handbook of Set Theory. Vols. 1, 2, 3 (2010), Springer: Springer Dordrecht), 395-489 · Zbl 1198.03058
[5] Bledsoe, W., Neighborly functions, Proc. Am. Math. Soc., 3, 114-115 (1952) · Zbl 0046.40301
[6] Bouziad, A.; Troallic, J.-P., Lower quasicontinuity, joint continuity and related concepts, Topol. Appl., 157, 18, 2889-2894 (2010) · Zbl 1213.54019
[7] Calbrix, J.; Troallic, J.-P., Applications séparément continues, C. R. Acad. Sci. Paris Sér. A-B, 288, 13, A647-A648 (1979)
[8] Engelking, R., General Topology (1989), Heldermann Verlag: Heldermann Verlag Berlin · Zbl 0684.54001
[9] Giles, J.; Kenderov, P.; Moors, W.; Sciffer, S., Generic differentiability of convex functions on the dual of a Banach space, Pac. J. Math., 172, 2, 413-431 (1996) · Zbl 0852.46019
[10] Gruenhage, G., Generalized metric spaces, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 423-501
[11] Gruenhage, G., Generalized metric spaces and metrization, (Recent Progress in General Topology. Recent Progress in General Topology, Prague, 1991 (1992), North-Holland: North-Holland Amsterdam), 239-274 · Zbl 0794.54034
[12] Gruenhage, G., Generalized metrizable spaces, (Recent Progress in General Topology. III (2014), Atlantis Press: Atlantis Press Paris), 471-505 · Zbl 1314.54001
[13] Kempisty, S., Sur les fonctions quasi-continues, Fundam. Math., 19, 184-197 (1932) · JFM 58.0246.01
[14] Kenderov, P.; Kortezov, I.; Moors, W., Continuity points of quasi-continuous mappings, Topol. Appl., 109, 3, 321-346 (2001) · Zbl 1079.54509
[15] Kenderov, P.; Moors, W., Fragmentability and sigma-fragmentability of Banach spaces, J. Lond. Math. Soc. (2), 60, 1, 203-223 (1999) · Zbl 0953.46004
[16] Kenderov, P.; Moors, W., Fragmentability of groups and metric-valued function spaces, Topol. Appl., 159, 1, 183-193 (2012) · Zbl 1253.46034
[17] Levine, N., Semi-open sets and semi-continuity in topological spaces, Am. Math. Mon., 70, 36-41 (1963) · Zbl 0113.16304
[18] Maslyuchenko, V. K., Joint continuous functions with values in inductive limits, Ukr. Mat. Zh., 44, 3, 380-384 (1992) · Zbl 0778.54003
[19] Maslyuchenko, V. K., Hahn spaces and the Dini problem, Mat. Metodi Fiz.-Mekh. Polya. Mat. Metodi Fiz.-Mekh. Polya, J. Math. Sci. (N.Y.), 107, 1, 3577-3582 (2001), translation in:
[20] Maslyuchenko, V. K., Separately continuous mappings of several variables with values in \(σ\)-metrizable spaces, Nelīnīĭnī Koliv., 2, 3, 337-344 (1999) · Zbl 1006.54017
[21] Maslyuchenko, V. K.; Mykhailyuk, V. V.; Filipchuk, O. I., Joint continuity points of separately continuous mappings with values in the Nemytskii plane, Mat. Stud., 26, 2, 217-221 (2006) · Zbl 1114.54012
[22] Maslyuchenko, V. K.; Mykhailyuk, V. V.; Filipchuk, O. I., Joint continuity of KhC-functions with values in Moore spaces, Ukr. Mat. Zh., 60, 11, 1539-1547 (2008) · Zbl 1199.54100
[23] Maslyuchenko, V. K.; Mykhailyuk, V. V.; Shishina, O. I., Joint continuity of horizontally quasicontinuous mappings with values in \(σ\)-metrizable spaces, Mat. Metodi Fiz.-Mekh. Polya, 45, 1, 42-46 (2002) · Zbl 1073.54008
[24] Maslyuchenko, V. K.; Mykhailyuk, V. V.; Sobchuk, O. V., Inverse problems in the theory of separately continuous functions, Ukr. Mat. Zh., 44, 9, 1209-1220 (1992) · Zbl 0791.54018
[25] Maslychenko, V. K.; Myronyk, O. D., Joint continuity of mappings with values in various generalizations of metrziable spaces, (All-Ukrainian Conf. “Modern Problems of Probability Theory and Mathematical Analysis”. All-Ukrainian Conf. “Modern Problems of Probability Theory and Mathematical Analysis”, 20-26 February 2012, Vorokhta, Ivano-Frankivsk (2012)), 5-6
[26] Maslyuchenko, V. K.; Nesterenko, V. V., Joint continuity and quasicontinuity of horizontally quasicontinuous mappings, Ukr. Mat. Zh.. Ukr. Mat. Zh., Ukr. Math. J., 52, 12, 1952-1955 (2000), translation in: · Zbl 0967.54017
[27] Maslyuchenko, V.; Nesterenko, V., A new generalization of Calbrix-Troallic’s theorem, Topol. Appl., 164, 162-169 (2014) · Zbl 1292.54008
[28] Michael, E., \( \aleph_0\)-spaces, J. Math. Mech., 15, 983-1002 (1966) · Zbl 0148.16701
[29] Moors, W.; Giles, J., Generic continuity of minimal set-valued mappings, J. Aust. Math. Soc. A, 63, 2, 238-262 (1997) · Zbl 0912.46017
[30] Myronyk, O., Stratifiable, Semistratifiable Spaces and Separately Continuous Mappings (2015), Ph.D. Thesis, Chernivtsi · Zbl 1363.54020
[31] Neubrunn, T., Quasi-continuity, Real Anal. Exch., 14, 2, 259-306 (1988/1989) · Zbl 0679.26003
[32] Piotrowski, Z., Separate and joint continuity, Real Anal. Exch., 11, 2, 293-322 (1985/1986) · Zbl 0606.54009
[33] Piotrowski, Z., Separate and joint continuity. II, Real Anal. Exch., 15, 1, 248-258 (1989/1990) · Zbl 0702.54009
[34] Piotrowski, Z., Separate and joint continuity in Baire groups, Tatra Mt. Math. Publ., 14, 109-116 (1998) · Zbl 0938.22001
[35] Ribarska, N., Internal characterization of fragmentable spaces, Mathematika, 34, 2, 243-257 (1987) · Zbl 0645.46017
[36] Sakai, K., Geometric Aspects of General Topology (2013), Springer: Springer Tokyo · Zbl 1280.54001
[37] Todorčević, S., Trees and linearly ordered sets, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 235-293 · Zbl 0557.54021
[38] Vaughan, J., Small uncountable cardinals and topology, (Open Problems in Topology (1990), North-Holland: North-Holland Amsterdam), 195-218
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