# Functionally Separable Subalgebra of C(X)

Document Type : Original Paper

Author

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

Abstract

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.

Keywords

Main Subjects

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### History

• Receive Date: 22 October 2020
• Revise Date: 16 January 2021
• Accept Date: 01 February 2021