Locally invariant topological groups and semidirect products

Authors:
R. W. Bagley and J. S. Yang

Journal:
Proc. Amer. Math. Soc. **93** (1985), 139-144

MSC:
Primary 22A05; Secondary 22C05

MathSciNet review:
766544

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider topological groups which have arbitrarily small invariant neighborhoods of the identity and those topological groups which admit continuous monomorphisms into such groups. We establish conditions under which the two corresponding classes of groups coincide. We apply these results to semidirect products. Since we do not assume local compactness in general, we use the symbol "[Sn]" rather than "[SIN]" for the class of groups with small invariant neighborhoods and the symbol "[In]" for those embeddable in Sn groups. We denote by "[N]" those groups with the property: If is a net in which converges to the identity and is any net such that converges, then this net also converges to the identity. We also define a class of topological groups we term groups. The following are corollaries of our general results: (1) If is locally compact, is compact and is an groups, then is an Sn group. (2) If is a locally connected compact group, is an Sn group, and if the semidirect product is an group, then is an Sn group. (3) If is an Sn group for every compact group , then every open subgroup of is of finite index.

**[1]**R. W. Bagley and K. K. Lau,*Semidirect products of topological groups with equal uniformities*, Proc. Amer. Math. Soc.**29**(1971), 179–182. MR**0274645**, 10.1090/S0002-9939-1971-0274645-8**[2]**Siegfried Grosser and Martin Moskowitz,*Compactness conditions in topological groups*, J. Reine Angew. Math.**246**(1971), 1–40. MR**0284541****[3]**Edward T. Ordman and Sidney A. Morris,*Almost locally invariant topological groups*, J. London Math. Soc. (2)**9**(1974/75), 30–34. MR**0364528****[4]**R. T. Ramsay,*Groups with equal uniformities*, Canad. J. Math.**21**(1969), 655–659. MR**0245718**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
22A05,
22C05

Retrieve articles in all journals with MSC: 22A05, 22C05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766544-X

Keywords:
Invariant neighborhoods of the identity,
nets,
semidirect products of topological groups,
locally compact groups,
compact groups

Article copyright:
© Copyright 1985
American Mathematical Society