A. R. Olfati
On a question of $C_c(X)$

Comment.Math.Univ.Carolin. 57,2 (2016) 253-260.

Abstract:In this short article we answer the question posed in Ghadermazi~M., Karamzadeh O.A.S., Namdari M., {\it On the functionally countable subalgebra of $C(X)$}, Rend. Sem. Mat. Univ. Padova {\bf 129} (2013), 47--69. It is shown that $C_c(X)$ is isomorphic to some ring of continuous functions if and only if $\upsilon_0 X$ is functionally countable. For a strongly zero-dimensional space $X$, this is equivalent to say that $X$ is functionally countable. Hence for every $P$-space it is equivalent to pseudo-$\aleph_0$-compactness.

Keywords: zero-dimensional space; strongly zero-dimensional space; $\mathbb{N}$-compact space; Banaschewski compactification; character; ring homomorphism; functionally countable subring; functional separability

DOI: DOI 10.14712/1213-7243.2015.161
AMS Subject Classification: 54C30 54D35 46E25 54D60 54C40