On selective sequential separability of function spaces with the compact-open topology

Year 2019, Volume: 48 Issue: 6 , 1761 - 1766 , 08.12.2019

Alexander V. Osipov

https://doi.org/10.15672/HJMS.2018.635

https://izlik.org/JA68BF55ET

Abstract

For a Tychonoff space $X$, we denote by $C_k(X)$ the space of all real-valued continuous functions on $X$ with the compact-open topology. A subset $A\subset X$ is said to be sequentially dense in $X$ if every point of $X$ is the limit of a convergent sequence in $A$. A space $C_k(X)$ is selectively sequentially separable (in Scheepers' terminology: $C_k(X)$ satisfies $S_{fin}(\mathcal{S},\mathcal{S})$) if whenever $(S_n : n\in \mathbb{N})$ is a sequence of sequentially dense subsets of $C_k(X)$, one can pick finite $F_n\subset S_n$ ($n\in \mathbb{N}$) such that $\bigcup \{F_n: n\in \mathbb{N}\}$ is sequentially dense in $C_k(X)$. In this paper, we give a characterization for $C_k(X)$ to satisfy $S_{fin}(\mathcal{S},\mathcal{S})$.

Keywords

compact-open topology , function space , selectively sequentially separable , S1(S , S) , sequentially dense set , property 2 , property 4

References

• [1] A.V. Arhangel’skii, The frequency spectrum of a topological space and the classification of spaces, Soviet Math. Dokl. 13, 1186–1189, 1972.
• [2] A. Bella, M. Bonanzinga, and M. Matveev, Variations of selective separability, Topol. App. 156, 1241–1252, 2009.
• [3] A. Bella, M. Bonanzinga, and M. Matveev, Sequential+separable vs sequentially separable and another variation on selective separability, Cent. Eur. J. Math. 11 (3), 530–538, 2013.
• [4] A. Bella and C. Costantini, Sequential Separability vs Selective Sequential Separability, Filomat 29 (1), 121–124, 2015.
• [5] A. Caserta, G. Di Maio, Lj.D.R. Kočinac, and E. Meccariello, Applications of k-covers II, Topol. App. 153, 3277–3293, 2006.
• [6] P. Gartside, J.T.H. Lo, and A. Marsh, Sequential density, Topol. App. 130, 75–86, 2003.
• [7] G. Gruenhage and M. Sakai, Selective separability and its variations, Topol. App. 158 (12), 1352–1359, 2011.
• [8] Lj.D.R. Kočinac, Closure properties of function spaces, App. Gen. Top. 4 (2), 255– 261, 2003.
• [9] Lj.D.R. Kočinac, $\gamma$-sets, $\gamma_k$-sets and hyperspaces, Mathematica Balkanica 19, 109– 118, 2005.
• [10] Lj.D.R. Kočinac, Selection principles and continuous images, Cubo Math. J. 8 (2), 23–31, 2006.
• [11] S. Lin, C. Liu, and H. Teng, Fan tightness and strong Fréchet property of $C_k(X)$, Adv. Math. (Chinese) 23 (3), 234–237, 1994.
• [12] G.Di Maio, Lj.D.R. Kočinac, and E. Meccariello, Applications of k-covers, Acta Math. Sin. (English Series) 22 (4), 1151–1160, 2006.
• [13] G.Di Maio, Lj.D.R. Kočinac, and T. Nogura Convergence properties of hyperspaces, J. Korean Math. Soc. 44 (4), 845–854, 2007.
• [14] A.J. Marsh, Topology of function spaces, Doctoral Dissertation, University of Pittsburgh, 2004.
• [15] R.A. McCoy, Function spaces which are k-spaces, Topol. P. 5, 139–146, 1980.
• [16] N. Noble, The density character of functions spaces, Proc. Amer. Math. Soc. 42, 228–233, 1974.
• [17] A.V. Osipov, Different kinds of tightness of a funtional space, Tr. Inst. Mat. Mekh. (Russian), 22 (3), 192–199, 2016.
• [18] A.V. Osipov, Application of selection principles in the study of the properties of function spaces, Acta Math. Hungar. 154 (2), 362–377, 2018.
• [19] A.V. Osipov, Classification of selectors for sequences of dense sets of $C_p(X)$, Topology Appl. 242, 20–32, 2018.
• [20] A.V. Osipov, The functional characterizations of the Rothberger and Menger properties, Topology Appl. 243, 146–152, 2018.
• [21] A.V. Osipov, Classification of selectors for sequences of dense sets of Baire functions, submitted.
• [22] A.V. Osipov and S. Özçağ, Variations of selective separability and tightness in function spaces with set-open topologies, Topology Appl. 217, 38–50, 2017.
• [23] A.V. Osipov and E.G. Pytkeev, On sequential separability of functional spaces, Topology Appl. 221, 270–274, 2017.
• [24] B.A. Pansera and V. Pavlović, Open covers and function spaces, Matematički Vesnik 58, 57–70, 2006.
• [25] M. Sakai, k-Frechet-Urysohn Property of $C_k(X)$, Topol. App. 154 (7), 1516–1520, 2007.
• [26] G. Tironi and R. Isler, On some problems of local approximability in compact spaces, In: General Topology and its Relations to Modern Analysis and Algebra III, 443–446, Prague, August 30-September 3, 1971, Academia, Prague, 1972.
• [27] A. Wilansky, How separable is a space?, Amer. Math. Monthly 79 (7), 764–765, 1972.