On the spectral height of \(F\)-compact spaces
The motivation for the study of \(F\)-compact spaces, introduced in [\textit{A. V. Ivanov}, in: Mappings and functors, Work Collect., Moskva 1984, 31--40 (1984; Zbl 0565.54015)], comes from Fedorchuk's method of fully closed mappings -- see e.g. [\textit{V. V. Fedorchuk}, Russ. Math. Surv. 35, No. 3, 131--143 (1980; Zbl 0529.54008); translation from Usp. Mat. Nauk 35, No. 3, 112--121 (Russian) (1980)] and [\textit{V. V. Fedorchuk}, Fundam. Prikl. Mat. 9, No. 4, 105--235 (2003); translation in J. Math. Sci., New York 136, No. 5, 4201--4292 (2006; Zbl 1073.54010)]. A fully closed mapping is a continuous mapping \(f:X\rightarrow Y\) such that, for every \(y\in Y\) and every finite cover \(\{U_1,\dots,U_n\}\) of \(f^{-1}[\{y\}]\) by open subsets of \(X\), the set \(\{y\}\cup\bigcup_{i=1}^{n}f^{\#}U_i\) is open in \(Y\); here, \(f^{\#}A=\{y\in Y:f^{-1}[\{y\}]\subseteq A\}\) is the small image of \(A\) under \(f\). A compact Hausdorff space is said to be an \(F\)-compact space if it is the limit of a well-ordered continuous inverse system \(\{X_{\alpha},p^{\alpha'}_{\alpha}:\alpha,\alpha'<\beta\}\) of compact Hausdorff spaces such that \(|X_0|=1\) and, for each \(\alpha\) with \(\alpha+1<\beta\), the bonding mapping \(p^{\alpha+1}_{\alpha}\) is fully closed and all of its preimages \((p^{\alpha+1}_{\alpha})^{-1}[\{x\}]\) for \(x\in X_\alpha\) are metrizable. The spectral height of an \(F\)-compact space \(X\), denoted by \(sh(X)\), is defined as the least length \(\beta\) of an inverse system that satisfies the above conditions and has limit \(X\). It follows from this definition that, if \(\gamma\) is a limit ordinal, then no \(F\)-compact space \(X\) satisfies \(sh(X)=\gamma+1\). In this paper, the authors show that, for every ordinal \(\beta\in(0,\omega_1]\) that is not the successor of a limit ordinal, there is an \(F\)-compact space \(X\) with \(sh(X)=\beta\). The construction makes use of the method of resolutions, also due to Fedorchuk see e.g. [loc. cit.].
scientific article; zbMATH DE number 6197547
scientific article