The purpose of this note is to discuss the relationship between the concepts of approximate continuity and almost continuity of real functions of a real variable.

Contrary to the papers of T. Husain [1], [2], [3], we shall show that there is little relationship between these concepts, that approximate continuity does not imply almost continuity without additional restrictions and we shall give one such restriction.

A recent issue of the Bulletin contained a paper by Robillard [13] in which results from the theory of confounded factorial designs were used to obtain some lower bounds for Ramsey numbers. We shall derive, by more elementary methods, bounds which are much better than Robillard's in every example which he considered.

In this paper, an elementary proof is given of the following proposition:

Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such that

I is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

Let A be a closed linear operator with domain DA dense in a Banach space B; DA is also a Banach space under the graph norm. By we represent the space of continuous linear mappings from B to DA and R(λ; A) = (λI—A)-1 denotes the resolvent of A; λ ∊ C (complex plane).

It is a well-known trivial fact that for a given integer n there exists at most n — 2 pairwise orthogonal idempotent latin squares. In the following note we prove that for n a prime power there always exists n—2 such squares.

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

It is well known that the wreath product of two ordered groups is an ordered group. In [2] Fuchs asks if the same is true for O*-groups. Here we construct an example to show that the wreath product of an infinite cyclic group with a free metabelian group is not an O*-group.

This note is closely related, as far as methods are concerned, to [3]. In [3] Ponomarev established “In order for a regular space X to be Lindelöf, it is necessary and sufficient that for each open covering ω of the space X there exists an ω-mapping ƒ:X → Y onto some separable metric space Y.” It is the purpose of this note to show that if the word “countable (or finite)” is inserted in the proper place we can obtain an analogous characterization for normal countably paracompact (or normal) spaces.

The concept of p-space is quite recent. It was introduced by Arhangel'skii [2]. The definition of p-space given in [2] involves compactification of the space. In view of the interesting properties of p-spaces obtained in [2], Alexadroff [1] suggested a problem of finding a direct intrinsic definition (without appeal to compactification). The main aim of this note is to answer the above problem.

I am grateful to Dr. S. K. Kaul for his comments.

Let A be a finite set of integers {ai} and A + A denote {ai + aj} with p different members and A — A denote {ai—aj) with m different members, the interesting conjecture of Conway [1] states that p < m, unless A is symmetric.