The number of cozero-sets is an \(\omega\)-power (Q2639365)

The following are crucial results of the paper: If X is an infinite Tychonoff space then \(| {\mathcal R}(X)|^{\omega}=| {\mathcal R}(X)|\) and \(| {\mathcal R}(X)| =| {\mathcal C}{\mathcal Z}(X)| =| {\mathcal C}(X)|\), where \({\mathcal C}(X)\) denotes the set of all continuous real-valued functions on X, \({\mathcal C}{\mathcal Z}(X)=\{X-f^{- 1}(0): f\in {\mathcal C}(X)\}\) and \({\mathcal R}(X)=\{Int cl U:U\in {\mathcal C}{\mathcal Z}(X)\}\). As a corollary it is shown that \(| {\mathcal T}(X)^{\omega}=| {\mathcal T}(X)|\) for each infinite perfectly normal space X.