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Some new trends in the theory of continuous mappings. (Russian. English summary) Zbl 0626.54023

Continuous functions on topological spaces, Collect. sci. Works, Riga 1986, 5-35 (1986).
[For the entire collection see Zbl 0619.00015.]
The author introduces the notion of P-splittability. Let P be a class of topological spaces. A space X is called P-splittable or splittable over P if for every pair of disjoint subsets A,B\(\subset X\) there exist \(Y\in P\) and a mapping \(f: X\to Y\) such that \(f(A)\cap f(B)=\emptyset\). Among other results it is shown that if P is multiplicative and hereditary, then for every P-splittable space X of cardinality C there exists a continuous bijection \(f: X\to Y\in P\). Hence a p-paracompactum of cardinality \(\leq C\) which is splittable over the class of metric spaces is metrizable. Compacta splittable over the class of spaces with \(G_{\delta}\)-diagonals or splittable over the class of symmetrizable Hausdorff spaces are metrizable, too. Special attention is paid to cardinal functions of spaces which are splittable over some classes. For example, if a compactum X is splittable over the class of Hausdorff spaces of countable tightness, then \(t(X)\leq \aleph_ 0\); if a compactum X is splittable over the class of Hausdorff \(\aleph_ 0\)-monolithic spaces of countable tightness then X is Fréchet-Uryson and \(\aleph_ 0\)-monolithic.

MSC:

54C99 Maps and general types of topological spaces defined by maps
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54E35 Metric spaces, metrizability

Citations:

Zbl 0619.00015