It is shown that a non-discrete locally compact Hausdorff group has each of its proper closed subgroups finite (respectively, discrete) if and only if it is topologically isomorphic to the circle group (respectively, the circle group or the group of real numbers).

In this brief note, we prove the following: Let R be a semi-primitive ring. Suppose that for each pair x, y ε R there exist positive integers m = m (x,y) and n = n (x,y) such that either [xm,(xy) n − (yx) n] = 0 or [xm,(xy) n + (yx) n] = 0. Then R is commutative.

In this paper we employ quasi-continuous multifunctions and introduce almost Baire class 1 multifunctions in order to generalize a theorem of Kuratowski and also to answer a question posed by him concerning Baire class 1 multifunctions. We also show that certain multifunctions can be decomposed into mutually singular multifunctions.

The object of this paper is to generalize the classic theorems of Eilenberg and Debreu on the existence of continuous order-preserving transformations on ordered topological spaces and to prove them in a different way. The proof of the theorems is based on Nachbin's generalization to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.

The concept of dependence of subgraphs of a plane graph is defined, as a measure of how much they overlap. It is shown that if M is a 3-connected plane graph, then the number of copies of M in a plane graph which are dependent on a given copy is bounded above by a constant c (M). The number of copies of M in any n-vertex plane graph is at most nc (m).

We give a new proof of the hypercentre theorem of Herstein.

In [1], Herstein has defined the hypercentre of a ring R as follows:

Herstein has proved:

THEOREM. If R is a ring without non-zero nil ideals, thenT (R) = Z (R).