zbMATH — the first resource for mathematics

Continuity, boundedness, connectedness and the Lindelöf property for topological groups. (English) Zbl 0724.22003
Let G be a locally compact abelian (LCA) group and \(G^+\) the group G equipped with the weakest topology such that each character of G remains continuous. The purpose of this paper is to compare properties of the topological groups G and \(G^+\). For instance, the following results are shown: (1) If G and H are LCA groups, then a homomorphism \(\Phi: G\to H\) is continuous if and only if \(\Phi: G^+\to H^+\) is continuous. (2) G is zero-dimensional if and only if so is \(G^+\). (3) The connected components of the neutral element of G and \(G^+\) are equal (4) \(F\subset G\) is Lindelöf or functionally bounded if and only if F has this property as subspace of \(G^+\). Some of these results remain valid for a larger class of topological groups than that consisting of all LCA groups. As the author points out, some of the results proved in the paper are well-known.
Reviewer: M.Voit (München)

22B05 General properties and structure of LCA groups
54C05 Continuous maps
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D05 Connected and locally connected spaces (general aspects)
Full Text: DOI
[1] Armacost, D.L., The structure of locally compact abelian groups, () · Zbl 0358.22001
[2] Comfort, W.W., Topological groups, (), 1143-1263 · Zbl 0604.22002
[3] Comfort, W.W.; Trigos, F.J., Theorem of Glicksberg, Abstracts amer. math. soc., 9, 420-421, (1988), (=Abstract #88T-22-195).
[4] de Vries, J.; Hus̆ek, M., Preservation of products by functors close to reflectors, Topology appl., 27, 171-189, (1987) · Zbl 0637.18002
[5] Dikranjan, D.N.; Prodanov, I.R.; Stoyanov, L.N., Topological groups, ()
[6] Engelking, R., General topology, () · Zbl 0684.54001
[7] Gillman, L.; Jerison, M., Rings of continuous functions, () · Zbl 0151.30003
[8] Glicksberg, I., Uniform boundedness for groups, Canad. math., 14, 269-276, (1962) · Zbl 0109.02001
[9] Hewitt, E.; Ross, K.A., Abstract harmonic analysis, volume 1, () · Zbl 0837.43002
[10] Heyer, H., Dualität lokalkompakter gruppen, () · Zbl 0202.14003
[11] Holm, P., On the Bohr compactification, Math. ann., 156, 34-46, (1964) · Zbl 0121.03705
[12] Hughes, R., Compactness in locally compact groups, Bull. amer. math. soc., 79, 122-123, (1973) · Zbl 0263.22006
[13] Kaplan, S., Extensions of Pontryagin duality I: infinite products, Duke math. J., 15, 649-658, (1948) · Zbl 0034.30601
[14] LaMartin, W.F., On the foundations of k-group theory, Dissertationes math., CXLVI, (1977) · Zbl 0394.22001
[15] Moran, W., On almost periodic compactifications of locally compact groups, J. London math. soc., 3, 2, 507-512, (1971) · Zbl 0229.22008
[16] Noble, N., κ-groups and duality, Trans. amer. math. soc., 151, 551-561, (1970) · Zbl 0229.22012
[17] D. Remus and F.J. Trigos-Arrieta, Remarks on a paper of Venkataraman (title tentative), Manuscript in preparation.
[18] Tkachenko, M.G., Boundedness and pseudocompactness in topological groups, Mat. zametki, Math. notes, 41, 229-231, (1987), English Translation · Zbl 0631.22005
[19] Trigos-Arrieta, F.J., Pseudocompactness on groups, Proceedings of the northeast conference on topology and its applications, (1989), to appear. · Zbl 0777.22003
[20] van Douwen, E., The maximal totally bounded group topology on G and the biggest minimal G-space, for abelian groups G, Topology appl., 34, 69-91, (1990) · Zbl 0696.22003
[21] Venkataraman, R., Compactness in abelian topological groups, Pacific J. math., 57, 591-595, (1975) · Zbl 0308.22009
[22] Willard, S., General topology, (1970), Addison Wesley Reading, MA · Zbl 0205.26601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.