It is shown that any hereditary lindel\"{o}f scattered space is functionally countable.Spaces $X$ such that $L_{cc}(X)$ is regular (von Neumann) are characterized and it is shown that $\aleph_0$-selfinjectivity and regularity of $L_{cc}(X)$ coincide.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we present a new subring of $C(X)$ that contains the subring $C_c(X)$, the set of all continuous functions with countable image.
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Key Findings & Data:Let $L_{cc}(X)=\{ f\in C(X)\,:\, |X\backslash C_f|\leq \aleph_0 \}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq \aleph_0$.
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Conclusion & Impact:It is shown that any hereditary lindel\"{o}f scattered space is functionally countable.Spaces $X$ such that $L_{cc}(X)$ is regular (von Neumann) are characterized and it is shown that $\aleph_0$-selfinjectivity and regularity of $L_{cc}(X)$ coincide.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:functionally countable space · uncountable-open space · socle
Abstract
In this paper, we present a new subring of $C(X)$ that contains the subring $C_c(X)$, the set of all continuous functions with countable image. Let $L_{cc}(X)=\{ f\in C(X)\,:\, |X\backslash C_f|\leq \aleph_0 \}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq \aleph_0$. We observe that $L_{cc}(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$. It is shown that any hereditary lindel\"{o}f scattered space is functionally countable.Spaces $X$ such that $L_{cc}(X)$ is regular (von Neumann) are characterized and it is shown that $\aleph_0$-selfinjectivity and regularity of $L_{cc}(X)$ coincide.
MEHRİ, R., MOHAMADİAN, R. (2022). On the locally countable subalgebra of C(X) whose local domain is cocountable. Hacettepe Journal of Mathematics and Statistics, Cilt: 46 - Sayı: 6, 1053–1068. https://ajindex.com/articles/529931
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