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

Authors : Alexander V OSİPOV

Pages : 1761-1766

Doi:10.15672/HJMS.2018.635

View : 23 | Download : 11

Publication Date : 2019-12-08

Article Type : Research Paper

Abstract :For a Tychonoff space $X$, we denote by $C_kinsert ignore into journalissuearticles values(X);$ 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 $C_kinsert ignore into journalissuearticles values(X);$ is selectively sequentially separable insert ignore into journalissuearticles values(in Scheepers` terminology: $C_kinsert ignore into journalissuearticles values(X);$ satisfies $S_{fin}insert ignore into journalissuearticles values(\mathcal{S},\mathcal{S});$); if whenever $insert ignore into journalissuearticles values(S_n : n\in \mathbb{N});$ is a sequence of sequentially dense subsets of $C_kinsert ignore into journalissuearticles values(X);$, one can pick finite $F_n\subset S_n$ insert ignore into journalissuearticles values($n\in \mathbb{N}$); such that $\bigcup \{F_n: n\in \mathbb{N}\}$ is sequentially dense in $C_kinsert ignore into journalissuearticles values(X);$. In this paper, we give a characterization for $C_kinsert ignore into journalissuearticles values(X);$ to satisfy $S_{fin}insert ignore into journalissuearticles values(\mathcal{S},\mathcal{S});$.
Keywords : compact open topology, function space, selectively sequentially separable, S1 S, S, sequentially dense set, property 2, property 4