Functionally Separable Subalgebra of C(X)

Document Type : Original Paper


Department of Science, Petroleum Faculty of Ahvaz, Petroleum University of Technology, Ahvaz, Iran


The useful role of $C_c(X)$ in studying $C(X)$ motivated us to introduce and study the functionally separable subalgebra $C_{cd}(Y)$ of $C(X)$. Let $Y$ be a dense subset of $X$, $C_{cd}(Y)={fin C(X): |f(Y)|leq {aleph}_0}$. Clearly, $C_c(X)subseteq C_{cd}(Y)subseteq C(X)$ and $C_{cd}(Y)$ behaves like $C(X)$ and $C_c(X)$ in more properties. If $X$ is a functionally countable or separable space then $C_{cd(Y)=C(X)$, in this case $X$ is called functionally separable space. Whenever $X$ is pseudocompact and $beta X$ is separable, then each $fin C(X)$ is countable on a dense subset of $X$. Conversely, if each $fin C(X)$ is countable on a dense subset of $X$ and each $G_{delta}$‌-set has nonempty interior, then $C(X)=C_c(X)$. ‌ Locally functionally separable subalgebra of $C(X)$ is denoted by $C_{cod}(X)$ where $C_{cod}(X)={fin C(X) : |f(Y)|leq aleph_0 , text{~~for some open dense subset $Y$ of $X$}}$, clearly $C_{cod}(X)subseteq L_c(X)$. For a locally compact and pseudocompact space $X$, $C_{cod}X)=C(X)$ if and only if $C_{cod}(beta X)=C(beta X)$. We introduce $z_{cod}$-ideals in $C_{cod}(X)$ and trivially observe that most of the facts related to $z$-ideals are extendable to $z_{cod}$-ideals.


Main Subjects

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