Bitopological realcompactness (Q2565560)

For the construction of realcompact pairwise extensions of a bitopological space, the authors consider strong interrelations between topologies, introduced by them in [Topology Appl. 42, 1--16 (1991; Zbl 0784.54033)]. Namely, a bitopological space \((X,\tau_1,\tau_2)\) has the property that \(\tau_1\) is a fine cotopology of \(\tau_2\) if (C0) \(\tau_1\) is a \(T_1\)-topology and \(\tau_1\subset \tau_2\); (C1) \(\tau_1\) is a completely regular with respect to \(\tau_1\) in the sense of \textit{E.~P.~Lane} [Proc. Lond. Math. Soc. (3) 17, 241--256 (1967; Zbl 0152.21101)]; furthermore, there are closed subbases \({\mathcal B}_1\) and \({\mathcal B}_2\) of \(\tau_1\) and \(\tau_2\), respectively, such that (S0) \({\mathcal B}_2\) is subbase-\(R_0\) with respect to \({\mathcal B}_1\), that is for every \(T\in {\mathcal B}_2\) and \(x\,\overline{\in}\,T\) there is an \(S\in {\mathcal B}_2\) such that \(x\in S\) and \(S\cap T=\varnothing\); (S1) the pair \(({\mathcal B}_1,{\mathcal B}_2)\) is pairwise normal in the sense of \textit{M.~J.~Saegrove} [J. Lond. Math. Soc. (2) 7, 286--290 (1973; Zbl 0266.54009)]; and, finally, (F) for every finite cover of \(X\) with elements of \(\{X\setminus S: S\in {\mathcal B}_1\}\) there is a finite refinement with members of \({\mathcal B}_2\); (C) for every countable cover of \(X\) with elements of \(\{X\setminus S:\;S\in {\mathcal B}_1\}\) there is a countable refinement with members of \({\mathcal B}_2\). But, by Theorem 3.1 from M.~J.~Saegrove [loc. cit.], (S0) and (S1) entail (C1). On this base the authors prove that if for a bitopological space \((X,\tau_1,\tau_2)\) there are closed subbases \({\mathcal B}_1\) and \({\mathcal B}_2\) of \(\tau_1\) and \(\tau_2\), respectively, such that the conditions (C0), (S0), (S1), (F) and (C) are satisfied, then there exists a pairwise embedding \(e: (X,\tau_1,\tau_2)\to (\widehat{X},\widehat{\tau}_1,\widehat{\tau}_2)\) such that \((\widehat{X},\widehat{\tau}_1)\) is realcompact with respect to \(\widehat{{\mathcal B}}_1\) and \(\widehat{{\mathcal B}}_2\), where \(\widehat{{\mathcal B}}_1\) and \(\widehat{{\mathcal B}}_2\) denote the families of closures in \((\widehat{X},\widehat{\tau}_2)\) of the elements of \({\mathcal B}_1\) and \({\mathcal B}_2\). It is necessary to note that the notion of bitopological realcompactness is derived from the characterization of topological realcompactness by \textit{I. van der Slot} [Fundam. Math. 67, 255--263 (1970; Zbl 0193.22803)], and is formulated as follows: a bitopological space \((X,\tau_1,\tau_2)\) with closed subbases \({\mathcal B}_1\) and \({\mathcal B}_2\) of \(\tau_1\) and \(\tau_2\), respectively, such that (C0), (C1), (F) and (C) are satisfied with respect to \({\mathcal B}_1\) and \({\mathcal B}_2\) is realcompact if every maximal centered system of \({\mathcal B}_1\) with the countable intersection property has a non-empty intersection.