The classical ring of quotients of \(C_c(X)\)
The ring of continuous functions of countable range on the space \(X\) is denoted by \(C_{c}(X)\). This very interesting concept was first introduced by Karamzadeh and Ghadermazi in [\textit{M. Ghadermazi} et al., Rend. Semin. Mat. Univ. Padova 129, 47--69 (2013; Zbl 1279.54015)].NEWLINENEWLINEThe classical ring of quotients of \(C(X)\) is denoted by \(q_{c}(X)\). In [\textit{M. Namdari} and \textit{A. Veisi}., Int. Math. Forum 7, No. 9--12, 561--571 (2012; Zbl 1252.54014)], the authors showed that \(q_{c}(X)=\lim _{_{O\in \mathcal{O(X)}}}C_{c}(O)\), where \(\mathcal{O(X)} =\{O\subseteq X: O\) is a dense cozeroset of \(X\}\).NEWLINENEWLINEIn the paper under review, Bhattacharjee, Knox and McGovern show that this classification need not be true for a zero-dimensional non-strongly zero-dimensional space (see Theorem 3.8) and they give a correct characterization of \(q_{c}(X)\). It is proved that \(q_{c}(X)=\lim _{_{U\in \mathcal{O}_{\sigma}(X)}} C_{c}(U)\), where \(\mathcal{O_{\sigma}(X)}=\{K\subseteq X: K\) is a dense \(\sigma\) clopen set of \(X\}\).
scientific article; zbMATH DE number 6353144
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