On products of quasi-\(F\)-compacta (Q1985624)

All maps here are continuous and surjective and all spaces are compact Hausdorff (=compacta). In this setting, a map \(f:X\rightarrow Y\) is \textit{fully closed} if for any disjoint closed sets \(F_1,F_2\subset X\) we have \(|f(F_1)\cap f(F_2)| < \omega\). The author calls \(f\) \textit{almost fully closed} if \(|f(F_1)\cap f(F_2)| \le \omega\). (We note that the term \textit{almost fully closed map} was used by \textit{V. V. Fedorchuk} [Fundam. Prikl. Mat. 9, No. 4, 105--235 (2003; Zbl 1073.54010); translation in J. Math. Sci., New York 136, No. 5, 4201--4292 (2006)] with a different meaning.) In this paper the author defines and studies quasi-\(F\)-compact Hausdorff spaces (or quasi-\(F\)-compacta, for short). Let \(X\) be a compact Hausdorff space and \(\alpha\), \(\beta\) and \(\gamma\) be ordinals. \(X\) is called a \textit{quasi-\(F\)-compactum} if it is the inverse limit of a well-ordered continuous inverse system \(S = \{X_\alpha,\pi^\alpha_\beta:\alpha,\beta\in\gamma\}\) such that \(X_0\) is a point, all neighboring projections \(\pi^{\alpha+1}_\alpha\), where \(\alpha+1\in\gamma\), are almost fully closed, and the inverse images \((\pi^{\alpha+1}_\alpha)^{-1}(x)\) are metrizable for every \(x\in X_\alpha\). An inverse system \(S\) satisfying these conditions is called \textit{quasi-\(F\)-system}. The \textit{spectral height} of a quasi-\(F\)-compactum \(X\) is the minimal length of a quasi-\(F\)-system \(S\) whose limit is \(X\). Clearly, every \(F\)-compactum is a quasi-\(F\)-compactum. Among other results the author shows that for any uncountable cardinal number \(\lambda\) there exists an \(F\)-compactum \(X\) of a spectral height \(\lambda\), all finite powers of which are \(F\)-compacta of the same spectral height; the product of a quasi-\(F\)-compactum (\(F\)-compactum) and a countable compactum is always a quasi-\(F\)-compactum (\(F\)-compactum); and the product of a quasi-\(F\)-compactum of spectral height 3 and an uncountable metrizable compactum is never a quasi-\(F\)-compactum of countable spectral height.