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- W3042136502 abstract "For a Hausdorff zero-dimensional topological space $X$ and a totally ordered field $F$ with interval topology, let $C_c(X,F)$ be the ring of all $F-$valued continuous functions on $X$ with countable range. It is proved that if $F$ is either an uncountable field or countable subfield of $mathbb{R}$, then the structure space of $C_c(X,F)$ is $beta_0X$, the Banaschewski Compactification of $X$. The ideals ${O^{p,F}_c:pin beta_0X}$ in $C_c(X,F)$ are introduced as modified countable analogue of the ideals ${O^p:pinbeta X}$ in $C(X)$. It is realized that $C_c(X,F)cap C_K(X,F)=bigcap_{pinbeta_0Xtexttt{textbackslash}X} O^{p,F}_c$, this may be called a countable analogue of the well-known formula $C_K(X)=bigcap_{pinbeta Xtexttt{textbackslash}X}O^p$ in $C(X)$. Furthermore, it is shown that the hypothesis $C_c(X,F)$ is a Von-Neumann regular ring is equivalent to amongst others the condition that $X$ is a $P-$space." @default.
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- W3042136502 date "2020-07-10" @default.
- W3042136502 modified "2023-09-27" @default.
- W3042136502 title "Ordered field valued continuous functions with countable range" @default.
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