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- W1855874989 abstract "<p><br />Let $C_c(X)={fin C(X) : |f(X)|leq aleph_0}$, $C^F(X)={fin C(X): |f(X)|<infty}$, and $L_c(X)={fin C(X) : overline{C_f}=X}$, where $C_f$ is the union of all open subsets $Usubseteq X$ such that $|f(U)|leqaleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered.<br />We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$.<br /> Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that $L_c(X)$ is a regular ring if and only if it is $aleph_0$-selfinjective.<br />We also determine spaces $X$ such that ${rm Soc}{big(}L_c(X){big)}=C_F(X)$ (resp., ${rm Soc}{big(}L_c(X){big)}={rm Soc}{big(}C_c(X){big)}$). It is proved that if $C_F(X)$ is a maximal ideal in $L_c(X)$, then $C_c(X)=C^F(X)=L_c(X)cong prodlimits_{i=1}^n R_i$, where $R_i=mathbb R$ for each $i$, and $X$ has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces $X$ for which $C_F(X)$ is a prime ideal in $L_c(X)$<br />are characterized and consequently for these spaces, we infer that $L_c(X)$ can not be isomorphic to any $C(Y)$. <br /><br /></p>" @default.
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- W1855874989 date "2015-10-01" @default.
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- W1855874989 title "On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)" @default.
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- W1855874989 doi "https://doi.org/10.4995/agt.2015.3445" @default.
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