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)


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


Zbl 0214.285
Full Text: DOI


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