Real Functions, Covers and Bornologies

Bukovský, Lev

doi:10.2478/tmmp-2021-0014

Abstract

The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

References

[1] BEER, G.—LEVI, S.: Strong uniform continuity, J. Math. Anal. Appl. 350 (2009), 568–589.10.1016/j.jmaa.2008.03.058
Search in Google Scholar Back to article
[2] BUKOVSKÝ, L.: Selection principle S1 and combinatorics of open covers, Topology Appl. 258 (2019), 239–252.10.1016/j.topol.2019.02.054
Search in Google Scholar Back to article
[3] BUKOVSKÝ, L.: Measurable functions and covering properties (to appear).
Search in Google Scholar Back to article
[4] BUKOVSKÝ, L.—HALEŠ, J.: QN-spaces, wQN-spaces and covering properties, Topology Appl. 154 (2007), 848–858.10.1016/j.topol.2006.09.008
Search in Google Scholar Back to article
[5] CASERTA, A.—DI MAIO, G.—HOLÁ, L.: Arceiá’s theorem and strong uniform convergence on bornologies, Topology Appl. 371 (2010), 384–392.
Search in Google Scholar Back to article
[6] ENGELKING, R.: General Topology. In: Monografie Matematyczne Vol. 60, PWN-Polish Scientific Publishers, Warszaw 1977. (Revised edition Heldermann Verlag, Berlin 1989.)
Search in Google Scholar Back to article
[7] GERLITS, F.—NAGY, Z.: Some properties of C(X), Part I, Topology Appl. 14 (1982), 151–161.10.1016/0166-8641(82)90065-7
Search in Google Scholar Back to article
[8] HALEŠ, J.: On Scheepers’ conjecture, Acta Univ. Carolin. Math. Phys. 46 (2005), no. 2, 27–31.
Search in Google Scholar Back to article
[9] HOLÁ, L’.—NOVOTNÝ, B.: Cardinal functions, bornologies and function spaces, Annali di Mathematica 193 (2014), 1319–1327.10.1007/s10231-013-0330-1
Search in Google Scholar Back to article
[10] JUST, W.—MILLER, A. W.—SCHEEPERS, M.—SZEPTICKI, P.J.: Combinatorics of open covers II, Topology Appl. 73 (1996), 241–266.10.1016/S0166-8641(96)00075-2
Search in Google Scholar Back to article
[11] OHTA, H.—SAKAI, M.: Sequences of semicontinuous functions accompanying continuous functions, Topology Appl. 156 (2009), 2683–2691.10.1016/j.topol.2009.07.017
Search in Google Scholar Back to article
[12] OSIPOV, A. V.: Classification of selectors for sequences of dense sets of Cp(X), Topology Appl. 242 (2018), 20–32.10.1016/j.topol.2018.04.010
Search in Google Scholar Back to article
[13] SAKAI, M.: Property C” and function spaces, Proc. Amer. Math. Soc. 104 (1988), 917–919.
Search in Google Scholar Back to article
[14] SAKAI, M.: Special subsets of reals characterising local properties of function spaces. In: Selection Principles and Covering Properties in Topology (Lj. D. R. Kočinac ed.), Quad. Mat. Vol. 18, Dept. Math., Seconda Univ. Napoli, Caserta, 2006 pp. 195–225.
Search in Google Scholar Back to article
[15] SCHEEPERS, M.: Combinatorics of open covers I: Ramsey theory, Topology Appl. 69 (1996), 31–62.10.1016/0166-8641(95)00067-4
Search in Google Scholar Back to article

Real Functions, Covers and Bornologies

Abstract

Paradigm

My account