On the role of finite, hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces.
The problems to which the reviewed paper is devoted originated from a celebrated theorem by \textit{A. Kurosch} [Compos. Math. 2, 471--476 (1935; Zbl 0012.32002)], according to which a compact Hausdorff space can be obtained as an inverse limit of finite \(T_0\)-spaces with continuous bonding maps. The method was developed by \textit{P. S. Alexandroff} and was applied by \textit{J. Flachsmeyer} [Math. Ann. 144, 253--274 (1961; Zbl 0118.17701)] in the theory of Hausdorff compactifications. The authors of the paper under review distinguish some special classes of finite \(T_0\)-spaces and some special kinds of bonding maps to obtain new special results in the area described above. The authors regard as one of the main results the Theorem 3.15 according to which the Hausdorff reflection of the inverse limit of normal finite \(T_0\)-spaces with continuous bonding maps is hereditarily disconnected. From the proof of this theorem the authors obtain as a corollary that the inverse limit of finite connected normal \(T_0\)-spaces with continuous bonding maps is normal and connected.
scientific article; zbMATH DE number 2037242