The purpose of this paper is to generalize some results in a recent paper by Tomiuk and the author ([11]).

For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].

The qualitative behaviour of control systems based on ordinary differential equations has been investigated with clarity and elegance using axiomatically defined General Control Systems. Here an attainablity set function, evolving in semigroup fashion, is the main entity of interest [1], [2], [3], [4].

In this paper our graphs will be finite, undirected, and without loops or multiple edges. We will denote the set of vertices of a graph G by V(G). If G is a graph and u, v∈V(G), then we will write u ∼ v to denote that u and v are adjacent and u ≁ v otherwise. If A ⊆ V(G), then we let N(A) = {u∈ V(G)|u ∼ a for each a ∈A}. However we write N(v) instead of N({v}). When there is no chance of confusion, we will not distinguish between a subset A ⊆ V(G) of vertices of G and the subgraph that it induces. We will denote the cardinality of a set A by |A|. The degree of a vertex v is δ(v) = |N(v)|. Any undefined terminology in this paper will generally conform with Behzad and Chartrand [1].

In [1] Atherton has asked whether the ideal topology (see [1]) is Hausdorff on every distributive lattice. He also gave some sufficient conditions (Kent's conditions cl and c3 [1]) under which the ideal topology on a complete distributive lattice is Hausdorff. In this paper we determine some more classes of lattices in which the ideal topology is Hausdorff and give examples to show that the classes determined are independent even in a complete distributive lattice. We also show by an example that Atherton's conditions are not necessary even in the case of a complete distributive lattice.

In [2] Hauptfleisch proved that if A, B, H, K are Abelian groups, φ:A → H and ψ:B → K are epimorphisms, then every central group extension G of H by K is homomorphic image of a central loop extension L of A by B. The aim of the present note is to prove (using almost the same argument as in [2])