Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
topological group; pseudocompact; Frechet-Urysohn; $G_\delta $-dense; $C$-embed\-ded; Moscow space; canonical uniform tightness; Hewitt completion; Rajkov completion; bounded set; extremally disconnected; normal space; $k_1$-space
topological group; pseudocompact; Frechet-Urysohn; $G_\delta $-dense; $C$-embed\-ded; Moscow space; canonical uniform tightness; Hewitt completion; Rajkov completion; bounded set; extremally disconnected; normal space; $k_1$-space
Summary:
A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is $C$-embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group $G$ and prove that every $G_\delta $-dense subspace $Y$ of a topological group $G$, such that the canonical uniform tightness of $G$ is countable, is $C$-embedded in $G$.
References:
[1] Arhangel'skii A.V.: Classes of topological groups. Russian Math. Surveys 36:3 (1981), 151-174. MR 0622722
[2] Arhangel'skii A.V.: Functional tightness, $Q$-spaces, and $\tau $-embeddings. Comment. Math. Univ. Carolinae 24:1 (1983), 105-120. MR 0703930
[3] Arhangel'skii A.V., Ponomarev V.I.: Fundamentals of General Topology in Problems and Exercises. D. Reidel Publ. Co., Dordrecht-Boston, Mass., 1984. MR 0785749
[4] Chigogidze A.C.: On $\kappa $-metrizable spaces. Russian Math. Surveys 37 (1982), 209-210. MR 0650791 | Zbl 0503.54012
[5] Comfort W.W., Ross K.A.: Pseudocompactness and uniform continuity in topological groups. Pacific J. Math. 16:3 (1966), 483-496. MR 0207886 | Zbl 0214.28502
[6] Comfort W.W., Trigos-Arrieta F.J.: Locally pseudocompact topological groups. Topology Appl. 62:3 (1995), 263-280. MR 1326826 | Zbl 0828.22003
[7] Engelking R.: General Topology. PWN, Warszawa, 1977. MR 0500780 | Zbl 0684.54001
[8] Glicksberg I.: Stone-Čech compactifications of products. Trans. Amer. Math. Soc. 90 (1959), 369-382. MR 0105667 | Zbl 0089.38702
[9] Hernandez S., Sanchis M.: $G_\delta $-open functionally bounded subsets in topological groups. Topology Appl. 53:3 (1993), 288-299. MR 1254873
[10] Hernandez S., Sanchis M., Tkachenko M.: Bounded sets in spaces and topological groups. preprint, 1997. Zbl 0979.54037
[11] Hušek M.: Productivity of properties of topological groups. Topology Appl. 44 (1992), 189-196. MR 1173257
[12] Michael E.: A quintuple quotient quest. Gen. Topol. and Appl. 2:1 (1972), 91-138. MR 0309045 | Zbl 0238.54009
[13] Pontryagin L.S.: Continuous Groups. Princeton Univ. Press, Princeton, N.J., 1956. Zbl 0659.22001
[14] Roelke W., Dierolf S.: Uniform Structures on Topological Groups and their Quotients. McGraw-Hill, New York, 1981.
[15] Tkachenko M.G.: Subgroups, quotient groups, and products of $R$-factorizable groups. Topology Proc. 16 (1991), 201-231. MR 1206464
[16] Tkachenko M.G.: Compactness type properties in topological groups. Czech. Math. J. 38 (1988), 324-341. MR 0946302 | Zbl 0664.54006
[17] Tkachenko M.G.: The notion of $o$-tightness and $C$-embedded subspaces of products. Topology Appl. 15 (1983), 93-98. MR 0676970 | Zbl 0509.54013
[18] Uspenskij V.V.: Topological groups and Dugundji spaces. Matem. Sb. 180:8 (1989), 1092-1118. MR 1019483
[19] de Vries J.: Pseudocompactness and the Stone-Čech compactification for topological groups. Nieuw Archef voor Wiskunde 23:3 (1975), 35-48. MR 0401978 | Zbl 0296.22003
Search
Partner of
