On the locally countable subalgebra of C(X) whose local domain is cocountable

Authors : R MEHRİ, R MOHAMADİAN

Pages : 1053-1068

View : 57 | Download : 8

Publication Date : 2017-12-01

Article Type : Research Paper

Abstract :In this paper, we present a new subring of $Cinsert ignore into journalissuearticles values(X);$ that contains the subring $C_cinsert ignore into journalissuearticles values(X);$, the set of all continuous functions with countable image. Let $L_{cc}insert ignore into journalissuearticles values(X);=\{ f\in Cinsert ignore into journalissuearticles values(X);\,:\, |X\backslash C_f|\leq \aleph_0 \}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|finsert ignore into journalissuearticles values(U);|\leq \aleph_0$. We observe that $L_{cc}insert ignore into journalissuearticles values(X);$ enjoys most of the important properties which are shared by $Cinsert ignore into journalissuearticles values(X);$ and $C_cinsert ignore into journalissuearticles values(X);$. It is shown that any hereditary lindel\`{o}f scattered space is functionally countable.Spaces $X$ such that $L_{cc}insert ignore into journalissuearticles values(X);$ is regular insert ignore into journalissuearticles values(von Neumann); are characterized and it is shown that $\aleph_0$-selfinjectivity and regularity of $L_{cc}insert ignore into journalissuearticles values(X);$ coincide.
Keywords : functionally countable space, uncountable open space, socle, locally c scattered space, aleph 0 selfinjective

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