Locales whose coz-complemented cozero sublocales have open closures (Q2310432)

In the paper under review, the authors study the pointfree counterpart of cloz-spaces, a class of spaces introduced by \textit{M. Henriksen} et al. [Diss. Math. 280, 31 p. (1989; Zbl 0719.54033)]. A cozero sublocale \(C\) of a completely regular locale \(L\) is \textit{coz-complemented} if there is a cozero sublocale \(D\) of \(L\) such that \(C\wedge D=C\cap D\) is the void sublocale \(\{1\}\) and \(C\vee D\) is dense in \(L\). Then \(L\) is said to be a \textit{cloz-locale} if every coz-complemented sublocale of \(L\) has an open closure. In particular, it turns out that a Tychonoff space is a cloz-space precisely when its locale of open sets is a cloz-locale. Cloz-locales are characterized in terms of F-quotients and certain sublocales of cloz-locales that inherit the property are identified. If the product of two cloz-locales is cloz then it is shown that each of the factors is cloz. Cloz-locales are also characterized in terms of some algebraic properties of their rings of real-valued continuous functions; namely, a locale \(L\) is a cloz-locale if and only if its ring of real continuous functions is what the authors call a \textit{nearly weak Baer ring}.