Classes SSGP(n)(n < omega) of topological groups are defined, and the class-theoretic inclusions SSGP(n) subseteq SSGP(n+1) subseteq m.a.p. are established and shown proper. These classes are investigated with respect to the properties normally studi
ed by topologists (products, quotients, passage to dense subgroups, and the like). In passing the authors establish the presence of the SSGP(1) or SSGP(2) property in many of the early examples in the literature of abelian m.a.p. groups.
Definition. Let $kappa$ be an infinite cardinal, let {X(i)} be a (not necessarily faithfully indexed) set of topological spaces, and let X be the product of the spaces X(i). The $kappa$-box product topology on X is the topology generated by those pro
ducts of sets U(i) for which (a) for each i, U(i) is open in X(i); and (b) U(i) = X(i) with fewer than $kappa$-many exceptions. (Thus, the usual Tychonoff product topology on X is the $omega$-box topology.) With emphasis on weight, density character, and Souslin number, the authors study and determine the value of several cardinal invariants on the space X with its $kappa$-box topology, in terms of the corresponding invariants of the individual spaces X(i). To the authors knowledge, this work is the first systematic study of its kind. Some of the results are axiom-sensitive, and some duplicate (and extend, and make precise) earlier work of Hewitt-Marczewski-Pondiczery, of Englking-Karlowicz, of Comfort-Negrepontis, and of Cater-Erdos-Galvin.
A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called locally precompact. Within the class of locally precompac
t groups, the authors classify those groups with the following topological properties: Dieudonne completeness; local realcompactness; realcompactness; hereditary realcompactness; connectedness; local connectedness; zero-dimensionality. They also prove that an abelian locally precompact group occurs as the quasi-component of a topological group if and only if it is precompactly generated, that is, it is generated algebraically by a precompact subset.
