Limit laws for wide varieties of topological groups. II (Q2716489)

[Part I by \textit{R. D. Kopperman}, \textit{M. W. Mislove}, \textit{S. A. Morris}, \textit{P. Nickolas}, \textit{V. Pestov} and \textit{S. Svetlichny} in ibid. 22, 307-328 (1996; Zbl 0892.22001).]NEWLINENEWLINENEWLINEA class of topological groups which is closed under the formation of subgroups, products and continuous homomorphic images is known as wide variety. For an infinite cardinal \(\kappa\), \(T(\kappa)\) provides an example of wide variety which contains a topological group iff each neighbourhood of its identity contains a normal subgroup of index strictly less than \(\kappa\). The paper contributes to the existing knowledge for the class of wide varieties \(T(\kappa)\), for infinite cardinals \(\kappa\). It is shown that the \(T(\kappa)\) are definable by a set of ``limit laws'' which may be defined in the following fashion: a ``limit law'' with respect to a directed set \(D\) and any set \(V\) is a formal expression \([\tau_d]\), where \(d\) runs through the elements of \(D\), and each \(\tau_d\) is a term in the first order theory of groups which has \(V\) as its set of variables.