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Factorization theorems for topological groups and their applications. (English) Zbl 0722.54039

A topological group G is called \({\mathbb{R}}\)-factorizable if for every continuous function f: \(G\to {\mathbb{R}}\) there exist a continuous homomorphism \(\pi\) : \(G\to H\) onto a topological group H of countable weight and a continuous function h: \(H\to {\mathbb{R}}\) with \(f=h_ 0\pi\). It is known that all pseudocompact groups are \({\mathbb{R}}\)-factorizable [W. W. Comfort and K. A. Ross, Pac. J. Math. 16, 483-496 (1966; Zbl 0214.285)]. In the paper under review other classes of \({\mathbb{R}}\)- factorizable topological groups are found. For example, all totally bounded and all Lindelöf topological groups are \({\mathbb{R}}\)-factorizable. The author proves also that if a compact space X is the continuous image of a dense subspace of a \(\sigma\)-compact topological group, then the weight and the tightness of X coincide. If in addition X is \(\aleph_ 0\)-monolothic, then X is metrizable. The latter results give answers to two problems of A. V. Arkhangel’skij.
Reviewer: L.Stoyanov (Sofia)

MSC:

54H11 Topological groups (topological aspects)
22A05 Structure of general topological groups
54C05 Continuous maps
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D30 Compactness

Citations:

Zbl 0214.285
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References:

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