Joint Continuity of Separately Continuous Mappings with Values in Completely Regular Spaces

Maslyuchenko, Volodymyr; Myronyk, Oksana; Filipchuk, Olha

doi:10.1515/tmmp-2017-0004

Abstract

We prove general theorems on quasi-continuity of mappings f : X₁ × ⋯ × X_n → Z with values in a completely regular space Z. As consequences, we obtain results on joint continuity of separately continuous functions of several variables involving the previous results of several authors.

References

[1] AARTS, J. M.—LUTZER, D. J.: Pseudo-completeness and the product of Baire spaces, Pacific J. Math. 48 (1973), 1–10.10.2140/pjm.1973.48.1
Search in Google Scholar Back to article
[2] BAIRE, R.: Sur les fonctions de variables réelles, Ann. Mat. Pura Appl. (4) 3 (1899), 1–123.10.1007/BF02419243
Search in Google Scholar Back to article
[3] BANAKH, T.: Quasicontinuous and separately continuous functions with values in Maslyuchenko spaces, Topology Appl. (to appear); preprint (http://arxiv.org/abs/1506.01661).
Search in Google Scholar Back to article
[4] BANAKH, T.: Quasicontinuous functions with values in Piotrowski spaces, preprint (http://arxiv.org/abs/1609.05482).
Search in Google Scholar Back to article
[5] BOUZIAD, A.—TROALLIC, J. P.: Lower quasicontinuity, joint continuity and related concepts, Topology Appl. 157 (2010), 2889–2894.10.1016/j.topol.2010.10.004
Search in Google Scholar Back to article
[6] CALBRIX, J.—TROALLIC, J. P.: Applications séparément continues, C.R. Acad. Sc. Paris. Séc. A. 288 (1979), 647–648.
Search in Google Scholar Back to article
[7] CEDER, J.: Some generalizations of metric spaces, Pacif. J. Math. 11 (1961), 105–126.10.2140/pjm.1961.11.105
Search in Google Scholar Back to article
[8] CHRISTENSEN, J. P. R.: Joint continuity of separately continuous functions, Proc. Amer. Math. Soc. 82 (1981), 455–461.10.1090/S0002-9939-1981-0612739-1
Search in Google Scholar Back to article
[9] ENGELKING, R.: General Topology. Mir, Moscow, 1986. (In Russian)
Search in Google Scholar Back to article
[10] FILIPCHUK, O. I.: On joint continuity of separately continuous mappings and its analogues with values in strongly σ-metrizable spaces, Nauk. Visn. Chernivets’kogo Univ., Mat. 336–337 (2007), 183–188. (In Ukrainian)
Search in Google Scholar Back to article
[11] FILIPCHUK, O. I.: Separately Continuous Mappings and Their Analogues with Values in Non-Metrizable Spaces. PhD Thesis, 2010. (In Ukrainian)
Search in Google Scholar Back to article
[12] FILIPCHUK, O. I.: Separately continuous functions with values in the Niemytski plane, in: Book of Abstracts, Int. Conf. dedicated to the 135th anniversary of Hans Hahn, Chernivtsi, Ukraine, 2014, pp. 205–206. (In Ukrainian)
Search in Google Scholar Back to article
[13] FLEISSNER, W. G.—KUNEN, K.: Barely Baire spaces, Fund. Math. 101 (1978), 229–240.10.4064/fm-101-3-229-240
Search in Google Scholar Back to article
[14] FROLÍK, Z.: Baire space and some generalizations of complete metric spaces, Czechoslovak Math. J. 11 (1961), 237–248.10.21136/CMJ.1961.100457
Search in Google Scholar Back to article
[15] HAHN, H.: Über Funktionen mehrerer Veränderlichen, die nach jeder einzelnen Veränderlichen stetig sind, Math. Zeitschr. 4 (1919), 306–313.10.1007/BF01203018
Open DOI Search in Google Scholar Back to article
[16] HAHN, H.: Theorie der Reellen Funktionen. 1. Band, J. Springer, Berlin, 1921.10.1007/978-3-642-52624-4_1
Search in Google Scholar Back to article
[17] HAHN, H.: Reelle Funktionen. 1. Teil. Punktfunktionen. Akademische Verlagsgesellschaft M. B. H., Leipzig, 1932.
Search in Google Scholar Back to article
[18] HAWORTH, R. C.—MCCOY, R. A.: Baire spaces, Dissertationes Math. (Rozprawy Mat.) 141 (1977), 1–77.
Search in Google Scholar Back to article
[19] HERNÁNDEZ, C.—MEDINA, L. R.—TKACHENKO, M.: Baire property in product spaces, Appl. Gen. Topol. 16 (2015), 1–13.10.4995/agt.2015.3439
Search in Google Scholar Back to article
[20] KARLOVA, O. O.—MASLYUCHENKO, V. K.—MYRONYK, O. D.: The Bing plane and separately continuous mappings, Mat. Stud. 38 (2012), 188–193. (In Ukrainian)
Search in Google Scholar Back to article
[21] KEMPISTY, S.: Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184–197.10.4064/fm-19-1-184-197
Search in Google Scholar Back to article
[22] KURATOWSKI, K.: Topology, Tom 1. Mir, Moscow, 1966. (In Russian)10.1016/B978-0-12-429201-7.50005-3
Search in Google Scholar Back to article
[23] MASLYUCHENKO, O. V.: Construction of ω-Primitives and Some Analogues of Compact Operators. Doctoral Thesis, 2012. (In Ukrainian)
Search in Google Scholar Back to article
[24] MASLYUCHENKO, V. K.: Separately continuous functions with values in strict inductive limits, in: XIV School on Theory of Operators in Functional Spaces, P. II, Novgorod, Russia, 1989. (In Russian)
Search in Google Scholar Back to article
[25] MASLYUCHENKO, V. K.: Separately continuous functions with values in inductive limits, Ukraïn. Mat. Zh. 44 (3) (1992), 380–384. (In Ukrainian)10.1007/BF01063133
Search in Google Scholar Back to article
[26] MASLYUCHENKO, V. K.: Separately Continuous Mappings and Köthe Spaces. Doctoral Thesis, 1999. (In Ukrainian)
Search in Google Scholar Back to article
[27] MASLYUCHENKO, V. K.: Separately continuous functions of many variables with values in σ-metrizable spaces, Nonlinear Oscil. 2 (1999), 337–344. (In Ukrainian)
Search in Google Scholar Back to article
[28] MASLYUCHENKO, V. K.: Hahn spaces and Dini’s problem, J. Math. Sci. (New York) 107 (2001), 3577–3582.10.1023/A:1011990123042
Search in Google Scholar Back to article
[29] MASLYUCHENKO, V. K.: First Types of Topological Vector Spaces. Ruta, Chernivtsi, 2002. (In Ukrainian)
Search in Google Scholar Back to article
[30] MASLYUCHENKO, V. K.: On separate and joint modifications of continuity, Mat. Stud. 25 (2006), 213–218. (In Ukrainian)
Search in Google Scholar Back to article
[31] MASLYUCHENKO, V. K.—FILIPCHUK, O. I.: Pointwise discontinuity of K_hK-functions with values in σ-metrizable spaces, Nauk. Visn. Chernivets’kogo Univ., Mat. 191–192 (2004), 103–106. (In Ukrainian)
Search in Google Scholar Back to article
[32] MASLYUCHENKO, V. K.—FILIPCHUK, O. I.: On the point discontinuity of K_hC-mappings on continuous curves, Nauk. Visn. Chernivets’kogo Univ., Mat. 314–315 (2006), 122–124. (In Ukrainian)
Search in Google Scholar Back to article
[33] MASLYUCHENKO, V. K.—FILIPCHUK, O. I.: Regular sequences of normed spaces and joint continuity of separately continuous mappings, Bukovinian Math. J. 3 (2015), 115–119. (In Ukrainian)
Search in Google Scholar Back to article
[34] MASLYUCHENKO, V. K.—MYKHAYLYUK, V. V.—FILIPCHUK, O. I.: Points of joint continuity of separately continuous mappings with values in the Niemytski plane, Mat. Stud. 26 (2006), 217–221. (In Ukrainian)
Search in Google Scholar Back to article
[35] MASLYUCHENKO, V. K.—MYKHAYLYUK, V. V.—FILIPCHUK, O. I.: Joint continuity of K_hC-functions with values in Moore spaces, Ukraïn. Mat. Zh. 60 (2008), 1539–1547. (In Ukrainian)
Search in Google Scholar Back to article
[36] MASLYUCHENKO, V. K.—MYKHAYLYUK, V. V.—SHYSHYNA, O. I.: Joint continuity of horizontal quasi-continuous mappings with values in σ-metrizable spaces, Mat. Metody i Fiz.-Mekh. Pólya 45 (2002), 42–46. (In Ukrainian)
Search in Google Scholar Back to article
[37] MASLYUCHENKO, V. K.—MYKHAYLYUK, V. V.—SOBCHUK, O. V.: On investigation of separately continuous functions, in: Proceedings of Intern. Math. Conf. dedicated to the anniv. of Hans Hahn, 1995, 192–246. (In Ukrainian)
Search in Google Scholar Back to article
[38] MASLYUCHENKO, V. K.—MYRONYK, O. D.: Separately continuous mappings and stratifiable spaces, Math. Visnyk NTSH 7 (2010), 111–121. (In Ukrainian)
Search in Google Scholar Back to article
[39] MASLYUCHENKO, V. K.—MYRONYK, O. D.: The Ceder product and stratifiable spaces, Bukovinian Math. J. 1 (2013), 107–112. (In Ukrainian)
Search in Google Scholar Back to article
[40] MASLYUCHENKO, V. K.—MYRONYK, O. D.: On continuity of KC-functions with values in the Ceder plane, Carpathian Math. Publ. 6 (2014), 329–336.10.15330/cmp.6.2.329-336
Search in Google Scholar Back to article
[41] MASLYUCHENKO, V. K.—MYRONYK, O. D.—FILIPCHUK, O. I.: Separately continuous mappings with values in completely regular spaces, in: Book of Abstracts, XI Summer School “Algebra, Topology and Analysis”, 2014, pp. 50–51. (In Ukrainian)
Search in Google Scholar Back to article
[42] MASLYUCHENKO, V. K.—NESTERENKO, V. V.: Joint continuity and quasi-continuity of horizontal quasi-continuous mappings, Ukraïn. Mat. Zh. 52 (2000), 1711–1714. (In Ukrainian)
Search in Google Scholar Back to article
[43] MASLYUCHENKO, V. K.—NESTERENKO, V. V.: Points of joint continuity and large oscillations, Ukraïn. Mat. Zh. 62 (2010), 791–800. (In Ukrainian)
Search in Google Scholar Back to article
[44] MASLYUCHENKO, V.—NESTERENKO, V.: A new generalization of Calbrix-Troallic’s theorem, Topology Appl. 164 (2014), 162–169.10.1016/j.topol.2013.12.014
Search in Google Scholar Back to article
[45] MOORS, W. B.: The product of a Baire with a hereditarily Baire metric space is Baire, Proc. Amer. Math. Soc. 134 (2006), 2161–2163.10.1090/S0002-9939-06-08389-4
Search in Google Scholar Back to article
[46] MYKHAYLYUK, V. V.: The Namioka property of KC-functions and Kempisty spaces, Topology Appl. 153 (2006), 2455–2461.10.1016/j.topol.2005.09.006
Search in Google Scholar Back to article
[47] MYKHAYLYUK, V. V.: Coordinate Method and Theory of Separately Continuous Mappings. Doctoral Thesis, 2009. (In Ukrainian)
Search in Google Scholar Back to article
[48] MYRONYK, O. D.: On separately continuous mappings with values in the Ceder plane, Bukovinian Math. J. 1 (2013), 100–105. (In Ukrainian)
Search in Google Scholar Back to article
[49] MYRONYK, O. D.: Separately continuous functions with values in the Ceder plane, in: Book of Abstracts, Int. Conf. dedicated to the 135th anniversary of Hans Hahn, Chernivtsi, Ukraine, 2014, pp. 131–132. (In Ukrainian)
Search in Google Scholar Back to article
[50] MYRONYK, O. D.: On the separately continuous functions with values in the Ceder plane, Bukovinian Math. J. 2 (2014), 173–176. (In Ukrainian)
Search in Google Scholar Back to article
[51] MYRONYK, O. D.: Stratifiable, Semistratifiable Spaces and Separately Continuous Mappings. PhD Thesis, 2015. (In Ukrainian)
Search in Google Scholar Back to article
[52] NEUBRUNN, T.: Quasi-continuity, Real Anal. Exch. 14 (1988–1989), 259–306.10.2307/44151947
Search in Google Scholar Back to article
[53] NESTERENKO, V. V.: Different Types of Quasi-Continuity and its Applications. PhD Thesis, 1999. (In Ukrainian)
Search in Google Scholar Back to article
[54] OXTOBY, J. C.: Cartesian products of Baire spaces, Fund. Math. 49 (1960/1961), 157–166.10.4064/fm-49-2-157-166
Search in Google Scholar Back to article
[55] PIOTROWSKI, Z.: Separate and joint continuity, Real Anal. Exch. 11 (1985), 293–322.10.2307/44151750
Search in Google Scholar Back to article
[56] PIOTROWSKI, Z.: Separate and joint continuity II, Real Anal. Exch. 15 (1989), 248–256.10.2307/44152002
Search in Google Scholar Back to article
[57] PIOTROWSKI, Z.: Mibu-type theorem, in: Classical Analysis, Proc. Inter. Symp., 1993, Radom, Poland, 1993, pp. 141–147.10.1017/S0308229600024661
Search in Google Scholar Back to article
[58] PIOTROWSKI, Z.: On the theorems of Y. Mibu and G. Debs on separately continuity, Inter. J. Math. and Math. Sci. 19 (1996), 495–500.10.1155/S0161171296000695
Open DOI Search in Google Scholar Back to article
[59] PIOTROWSKI, Z.—WALLER, R.: Baire and weakly Namioka spaces, Topology Appl. 159 (2012), 3294–3299.10.1016/j.topol.2012.07.006
Search in Google Scholar Back to article
[60] WESTON, J. O.: Some theorem on cluster sets, J. London Math. Soc. 33 (1958), 435–441.10.1112/jlms/s1-33.4.435
Open DOI Search in Google Scholar Back to article
[61] WHITE, H. E., JR.: Topological spaces that are α-favorable for a player with perfect information, Proc. Amer. Math. Soc. 50 (1975), 477–482.
Search in Google Scholar Back to article

Joint Continuity of Separately Continuous Mappings with Values in Completely Regular Spaces

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