×

Further applications of bornological covering properties in function spaces. (English) Zbl 1486.54038

A bornology \(\mathfrak{B}\) on a set \(X\) is a family of subsets of \(X\) that is closed under taking finite unions, is hereditary under inclusion and forms a cover of \(X\). This article is a follow up of the work done in the function space \(C(X)\) with respect to the topology \(\tau^{S}_{\mathfrak{B}}\) of strong uniform convergence on \(\mathfrak{B}\), carried out in [D. Chandra et al., Indag. Math., New Ser. 31, No. 1, 43–63 (2020; Zbl 1432.54019)] using the idea of strong uniform convergence [G. Beer and S. Levi, J. Math. Anal. Appl. 350, No. 2, 568–589 (2009; Zbl 1161.54003)] on a bornology. The authors present characterizations of various local properties of \((C(X),\tau^{S}_{\mathfrak{B}})\) such as countable fan tightness for finite sets, countable strong fan tightness for finite sets, Fréchet-Urysohn for finite sets etc. in terms of selection principles related to bornological covers of \(X\) (represented by a diagram describing various implications among them). Lastly, taking cue from [M. Scheepers, Fundam. Math. 152, No. 3, 231–254 (1997; Zbl 0884.90149); Quaest. Math. 22, No. 1, 109–130 (1999; Zbl 0972.91026)], the authors also investigate topological properties of sequences of dense and sequentially dense subsets of \((C(X),\tau^{S}_{\mathfrak{B}})\) and present a diagram describing implications among those selective properties of \((C(X),\tau^{S}_{\mathfrak{B}})\).

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C35 Function spaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arkhangel’skii, A. V., Hurewicz spaces, analytic sets and fan tightness of function spaces, Sov. Math. Dokl., 33, 396-399 (1986) · Zbl 0606.54013
[2] Arhangel’skii, A. V.; Isler, R.; Tironi, G., On pseudo-radial spaces, Comment. Math. Univ. Carol., 27, 137-154 (1986) · Zbl 0587.54007
[3] Arhangel’skii, A. V., Topological Function Spaces (1992), Kluwer · Zbl 0911.54004
[4] Bella, A.; Bonanzinga, M.; Matveev, M., Variations of selective separability, Topol. Appl., 156, 1241-1252 (2009) · Zbl 1168.54009
[5] Beer, G.; Levi, S., Pseudometrizable bornological convergence is Attouch-Wets convergence, J. Convex Anal., 15, 439-453 (2008) · Zbl 1173.54002
[6] Beer, G.; Levi, S., Strong uniform continuity, J. Math. Anal. Appl., 350, 568-589 (2009) · Zbl 1161.54003
[7] Bonanzinga, M.; Cammaroto, F.; Matveev, M., Projective versions of selection principles, Topol. Appl., 157, 874-893 (2010) · Zbl 1189.54016
[8] Bukovský, L.; Haleš, J., QN-spaces, wQN-spaces and covering properties, Topol. Appl., 154, 848-858 (2007) · Zbl 1117.54003
[9] Caserta, A.; Di Maio, G.; Kočinac, Lj. D.R.; Meccariello, E., Applications of k-covers II, Topol. Appl., 153, 3277-3293 (2006) · Zbl 1117.54034
[10] Caserta, A.; Di Maio, G.; Holá, Ĺ., Arzelà’s theorem and strong uniform convergence on bornologies, J. Math. Anal. Appl., 371, 384-392 (2010) · Zbl 1202.54004
[11] Caserta, A.; Di Maio, G.; Kočinac, Lj. D.R., Bornologies, selection principles and function spaces, Topol. Appl., 159, 7, 1847-1852 (2012) · Zbl 1253.54021
[12] Chandra, D.; Das, P.; Das, S., Applications of bornological covering properties in metric spaces, Indag. Math., 31, 43-63 (2020) · Zbl 1432.54019
[13] Chandra, D.; Das, P.; Das, S., Certain observations on selection principles from (a) bornological viewpoint, Quaest. Math. (2022), (to appear) · Zbl 1491.54022
[14] Engelking, R., General Topology, Sigma Ser. Pure Math. (1989), Heldermann: Heldermann Berlin · Zbl 0684.54001
[15] Gerlits, J.; Nagy, Zs., Some properties of \(C(X), I\), Topol. Appl., 14, 151-161 (1982) · Zbl 0503.54020
[16] Gruenhage, G.; Szeptycki, P. J., Fréchet-Urysohn for finite sets, Topol. Appl., 151, 238-259 (2005) · Zbl 1085.54016
[17] Hogbe-Nlend, H., Bornologies and Functional Analysis (1977), North-Holland: North-Holland Amsterdam · Zbl 0359.46004
[18] Juhász, I., Variations on tightness, Studia Sci. Math. Hung., 24, 179-186 (1989) · Zbl 0695.54005
[19] Just, W.; Miller, A. W.; Scheepers, M.; Szeptycki, P. J., The combinatorics of open covers (II), Topol. Appl., 73, 241-266 (1996) · Zbl 0870.03021
[20] Kočinac, Lj. D.R.; Scheepers, M., Function spaces and a property of Reznichenko, Topol. Appl., 123, 135-143 (2002) · Zbl 1011.54017
[21] Kočinac, Lj. D.R., Closure properties of function spaces, Appl. Gen. Topol., 4, 2, 255-261 (2003) · Zbl 1055.54007
[22] Kočinac, Lj. D.R.; Scheepers, M., Combinatorics of open covers (VII), Fundam. Math., 179, 131-155 (2003) · Zbl 1115.91013
[23] Kočinac, Lj. D.R., Selected results on selection principles, (Proc. Third Seminar on Geometry and Topology. Proc. Third Seminar on Geometry and Topology, Tabriz, Iran, July 15-17 (2004)), 71-104
[24] Kočinac, Lj. D.R., γ-sets, \( \gamma_k\)-sets and hyperspaces, Math. Balk., 19, 109-118 (2005) · Zbl 1086.54008
[25] Lin, S.; Liu, C.; Teng, H., Fan tightness and strong Fréchet property of \(C_k(X)\), Adv. Math., 23, 3, 234-237 (1994), (in Chinese), MR. 95e:54007, Zbl. 808.54012 · Zbl 0808.54012
[26] McCoy, R. A., Function spaces which are k-spaces, Topol. Proc., 5, 139-146 (1980) · Zbl 0461.54011
[27] McCoy, R. A.; Ntantu, I., Topological Properties of Spaces of Continuous Functions, Lecture Notes in Math., vol. 1315 (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0647.54001
[28] Noble, N., The density character of functions spaces, Proc. Am. Math. Soc., 42, 228-233 (1974) · Zbl 0278.54016
[29] Osipov, A. V., Classification of selectors for sequences of dense sets of \(C_p(X)\), Topol. Appl., 242, 20-32 (2018) · Zbl 1398.54031
[30] Osipov, A. V., Selection principles in function spaces with the compact-open topology, Filomat, 15, 5403-5413 (2018) · Zbl 1042.91546
[31] Osipov, A. V., On selective sequential separability of function spaces with the compact-open topology, Hacet. J. Math. Stat., 48, 6, 1761-1766 (2019) · Zbl 1488.54063
[32] Reznichenko, E.; Sipacheva, O., Fréchet-Urysohn type properties in topological spaces, groups and locally convex vector spaces, Mosc. Univ. Math. Bull., 54, 3, 33-38 (1999) · Zbl 0949.54005
[33] Sakai, M., Property \(C''\) and function spaces, Proc. Am. Math. Soc., 104, 917-919 (1988) · Zbl 0691.54007
[34] Sakai, M., Variations on tightness in function spaces, Topol. Appl., 101, 273-280 (2000) · Zbl 0955.54005
[35] Sakai, M., The sequence selection properties of \(C_p(X)\), Topol. Appl., 154, 552-560 (2007) · Zbl 1109.54014
[36] Sakai, M., Selective separability of Pixley Roy hyperspaces, Topol. Appl., 159, 1591-1598 (2012) · Zbl 1251.54015
[37] Sakai, M., The projective Menger property and an embedding of \(S_\omega\) into function spaces, Topol. Appl., 220, 118-130 (2017) · Zbl 1365.54020
[38] Scheepers, M., Combinatorics of open covers (I): Ramsey theory, Topol. Appl., 69, 31-62 (1996) · Zbl 0848.54018
[39] Scheepers, M., Combinatorics of open covers (III): \( C_p(X)\) and games, Fundam. Math., 152, 231-254 (1997) · Zbl 0884.90149
[40] Scheepers, M., Combinatorics of open covers (VI): selectors for sequences of dense sets, Quaest. Math., 22, 109-130 (1999) · Zbl 0972.91026
[41] Scheepers, M., Sequential convergence in \(C_p(X)\) and a covering property, East-West J. Math., 1, 2, 207-214 (1999) · Zbl 0976.54016
[42] Scheepers, M., Selection principles and covering properties in topology, Note Mat., 22, 2, 3-41 (2003/2004) · Zbl 1195.37029
[43] Tsaban, B., Some new directions in infinite combinatorial topology, (Bagaria, J.; Todorcevic, S., Set Theory. Set Theory, Trends Math. (2006), Birkhauser), 225-255 · Zbl 1113.54002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.