In this paper, we give a characterization for $C_k(X)$ to satisfy $S_{fin}(\mathcal{S},\mathcal{S})$.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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 $A$.
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Conclusion & Impact:In this paper, we give a characterization for $C_k(X)$ to satisfy $S_{fin}(\mathcal{S},\mathcal{S})$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:compact-open topology · function space · selectively sequentially separable
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 $S_{fin}(\mathcal{S},\mathcal{S})$.
Osipov, A. V. (2022). On selective sequential separability of function spaces with the compact-open topology. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 6, 1761–1766. https://ajindex.com/articles/530114
A.V. Arhangel’skii, The frequency spectrum of a topological space and the classification of spaces, Soviet Math. Dokl. 13, 1186–1189, 1972
A. Bella, M. Bonanzinga, and M. Matveev, Variations of selective separability, Topol. App. 156, 1241–1252, 2009
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
A. Bella and C. Costantini, Sequential Separability vs Selective Sequential Separability, Filomat 29 (1), 121–124, 2015
A. Caserta, G. Di Maio, Lj.D.R. Kočinac, and E. Meccariello, Applications of k-covers II, Topol. App. 153, 3277–3293, 2006
P. Gartside, J.T.H. Lo, and A. Marsh, Sequential density, Topol. App. 130, 75–86, 2003
G. Gruenhage and M. Sakai, Selective separability and its variations, Topol. App. 158 (12), 1352–1359, 2011
Lj.D.R. Kočinac, Closure properties of function spaces, App. Gen. Top. 4 (2), 255– 261, 2003