\(C(X)\) versus its functionally countable subalgebra
In the spirit of the study of rings of continuous functions, the authors study the interplay between some topological properties of zero-dimensional spaces $X$ and algebraic properties of the rings $C_c(X)$ of continuous real-valued functions on $X$ having countable image. They show, among other things, that two zero-dimensional spaces $X$ and $Y$ are homeomorphic if and only if $C_c(X)$ and $C_c(Y)$ are isomorphic. They characterize spaces $X$ for which every function in $C_c(X)$ actually has finite image. These are precisely the spaces $X$ for which $C_c(X)$ is von Neumann regular and each locally finite family of clopen subsets of $X$ is finite. A curious observation is that, whereas the socle of $C(X)$ is never a prime ideal in $C(X)$, there are instances when it is a prime ideal in $C_c(X)$. The authors actually characterize the spaces for which this is the case.
scientific article; zbMATH DE number 7040768
scientific article