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Journals / Hacettepe Journal of Mathematics and Statistics / 2019 / Volume: 48 Issue: 6

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

Alexander V. Osipov(Ural State University of Economics)0000-0002-6460-1943
Pages
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Publication Date
2019-12-08
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⚡ TL;DR(Too Long; Didn't Read)

In this paper, we give a characterization for $C_k(X)$ to satisfy $S_{fin}(\mathcal{S},\mathcal{S})$.

📌 3 Key Takeaways

🎯
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 $S_{fin}(\mathcal{S},\mathcal{S})$.

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Cite This Article

Osipov, A. V. (2019). On selective sequential separability of function spaces with the compact-open topology. Hacettepe Journal of Mathematics and Statistics, Volume: 48 Issue: 6. https://doi.org/10.15672/HJMS.2018.635

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