On the extension of Aleksandrov theorem to mappings
In the theory of continuous mappings ``one of the latest promotions, unexpected and giving new directions for development, is the transfer by B. A. Pasynkov of fundamental notions of general topology, such as compact-like notions, metrizability, cardinal invariants, dimensions, to the class of continuous mappings'' [\textit{P. S. Aleksandrov}, \textit{V. V. Fedorchuk} and \textit{V. I. Zajtsev}, Usp. Mat. Nauk 33, No. 3(201), 3-48 (1978; Zbl 0389.54001)]. The paper under review is a further promotion in this direction. In particular, it generalizes Aleksandrov's theorem to the class of continuous mappings: a space of weight \(\tau\) is inductively zero- dimensional iff it is embeddable in the Cantor cube \(D^\tau\). It should be noted that the role of the Cantor cube in the class of continuous mappings is played by the projection \(p : P \to Y\), where \(P = \mathbb{P}(Y, \{Z_\alpha = D\},\;\{O_\alpha\})\), \(\alpha \in A\), \(|A|= \tau\) and \(D\) being a two-point discrete space, is the partial product [\textit{B. A. Pasynkov}, Tr. Mosk. Mat. O.-va 13, 136-245 (1965; Zbl 0147.413)] and a mapping \(f : X \to Y\) is called embeddable in a mapping \(g: Z \to Y\) if there exists a homeomorphic embedding \(h : X \to Z\) such that \(f = g \circ h\).
scientific article; zbMATH DE number 732064