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Spaces that are projective with respect to classes of mappings. (English. Russian original) Zbl 0662.54007

Trans. Mosc. Math. Soc. 1988, 139-156 (1988); translation from Tr. Mosk. Mat. O.-va 50, 138-155 (1987).
Among the theorems which motivate paper under review let us mention the following one due to B. A. Pasynkov [Dokl. Akad. Nauk SSSR 175, 292-295 (1967; Zbl 0155.502)]: If a space X is Čech complete and f is a (continuous) mapping from X onto a paracompact space, then there exists a closed subset A of X such that \(f| A\) is perfect (i.e. closed and having compact point inverses) and has the same range as f; such a mapping f is called inductively perfect, being perfect if \(A=X\). The author introduces a system of notions in which a space X satisfying the conclusion of the mentioned above theorem is called inductively perfect with respect to the class of paracompact spaces. More generally, a (Tichonov) space X is called F-projective with respect to a given class of spaces, where F stand for a given class of mappings, if each mapping from X onto a space from the given class belongs to F; if a restriction of the mapping to a closed subset of X is needed, the space X is called inductively F-projective. Several classes of spaces are characterized as inductively F-projective with respect to appropriate F’s and appropriate classes of spaces. None the less in the most interesting case of Pasynkov’s theorem, the author gives an example showing that not only Čech complete spaces are inductively perfect. The author regards his research as a part of a program of Alexandrov to investigate spaces which can be mapped onto “good” spaces by means of “good” mappings, having in view those theorems the conclusions of which concern the mappings, not the spaces.
Reviewer: J.Mioduszewski

MSC:

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness
54E52 Baire category, Baire spaces

Citations:

Zbl 0155.502
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