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# Article

 Title: Pseudouniform topologies on $C(X)$ given by ideals (English) Author: Pichardo-Mendoza, Roberto Author: Tamariz-Mascarúa, Ángel Author: Villegas-Rodríguez, Humberto Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 54 Issue: 4 Year: 2013 Pages: 557-577 Summary lang: English . Category: math . Summary: Given a Tychonoff space $X$, a base $\alpha$ for an ideal on $X$ is called pseudouniform if any sequence of real-valued continuous functions which converges in the topology of uniform convergence on $\alpha$ converges uniformly to the same limit. This paper focuses on pseudouniform bases for ideals with particular emphasis on the ideal of compact subsets and the ideal of all countable subsets of the ground space. (English) Keyword: function space Keyword: topology of uniform convergence Keyword: ideal Keyword: uniformity Keyword: Lindelöf property Keyword: pseudouniform ideal Keyword: almost pseudo-$\omega$-bounded MSC: 54A10 MSC: 54A20 MSC: 54A25 MSC: 54C35 MSC: 54D20 MSC: 54E15 idZBL: Zbl 06373983 idMR: MR3125075 . Date available: 2013-10-01T21:19:06Z Last updated: 2016-01-04 Stable URL: http://hdl.handle.net/10338.dmlcz/143475 . Reference: [1] Arkhangel'skii A.V.: Topological Function Spaces.Mathematics and its Applications, 78, Kluwer Academic Publishers, Dordrecht, 1992 (translated from the Russian). MR 1144519 Reference: [2] Engelking R.: General Topology.second edition, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989; translated from the Polish by the author; MR 91c:54001. Zbl 0684.54001, MR 1039321 Reference: [3] Gul'ko S.P.: On properties of subsets of $\Sigma$-products.Soviet Math. Dokl. 18 (1977), 1438–1442. Reference: [4] Isiwata T.: On convergences of sequences of continuous functions.Proc. Japan Acad. 37 (1961), no. 1, 4–9. Zbl 0102.32201, MR 0141069 Reference: [5] Jech T.: Set Theory.The Third Milleniun Edition, revised and expanded, Springer Monographs in Mathematics, 3rd rev. ed. Corr. 4th printing, Springer, Berlin, 2003. Zbl 1007.03002, MR 1940513 Reference: [6] Kundu S., McCoy R.A.: Topologies between compact and uniform convergence on function spaces.Internat. J. Math. Math. Sci. 16 (1993), no. 1, 101–109. Zbl 0798.54020, MR 1200117, 10.1155/S0161171293000122 Reference: [7] Kunen K.: Set Theory. An Introduction to Independence Proofs.Studies in Logic and the Foundations of Mathematics, 102, North-Holland Publishing Co., Amsterdam, 1980. Zbl 0534.03026, MR 0597342 Reference: [8] McCoy R.A., Ntantu I.: Topological properties of spaces of continuous functions.Lecture Notes in Mathematics, 1315, Springer, Berlin, 1988. Zbl 0647.54001, MR 0953314 Reference: [9] Shakhmatov D.B.: A pseudocompact Tychonoff space all countable subsets of which are closed and $C^*$-embedded.Topology Appl. 22 (1986), no. 2, 139–144. MR 0836321, 10.1016/0166-8641(86)90004-0 Reference: [10] Todorčević S.: Trees and linearly ordered sets.Handbook of Set-Theoretic Topology, (K. Kunen and J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 235–293. Zbl 0557.54021, MR 0776625 Reference: [11] Turzanski M.: On generalizations of dyadic spaces.in Frolík Z. (ed.), Proceedings of the 17th Winter School on Abstract Analysis. Charles University, Praha, 1989, pp. 153–159. Zbl 0713.54040, MR 1046462 Reference: [12] Vaughan J.E.: Countably compact and sequentially compact spaces.Handbook of Set-Theoretic Topology, (K. Kunen and J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 569–602. Zbl 0562.54031, MR 0776631 .

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