It is proved that every metrizable topological space without isolated points is the union of a continuum or fewer nowhere dense subsets.

A generating function is derived for the number of transitions to the first passage through a particular vertex of a random walk on a doubly regular tournament. As an application of this result, we obtain a generating function for the number of solutions (q1, …, qk) of qi ≡ r (mod p), where p is a prime of the form 4l + 3 and the qi are quadratic residues modulo p.

Let G be a compact Lie group and V and W be linear G spaces. A study is made of the canonical stratification of some algebraic varieties that arise naturally in the theory of C∞ equivariant maps from V to W. The main corollary of our results is the equivalence of Bierstone's concept of “equivariant general position” with our own of “G transversal”. The paper concludes with a description of Bierstone's higher order conditions for equivariant maps in the framework of equisingularity sequences.

This paper produces new types of designs, called product designs, which prove extremely useful for constructing orthogonal designs. An orthogonal design of order 2t and type

is constructed. This design often meets the Radon bound for the number of variables.

We also show that all orthogonal designs of order 2t and type (a, b, c, d, 2t-a-b-c-d), with 0 < a + b + c + d < 2t, exist for t = 5, 6, and 7.

Perhaps the simplest example is the lexicographic sum of copies Zn of the integers indexed by the integers. If one performs the shift in the 0-indexed summand m0 → (m+1)0 while leaving all the other summands fixed, and follows this with the shift Zn → Zn+1 in the index set, then the element 00. will be sent on 11; whereas under these automorphisms performed in the other order, it will be sent on 01. This non-commutativity contradicts Theorem 9 in [1].

Dr Isidore Fleischer has shown by a very simple counterexample [1] that Theorem 9 (p. 30) of my paper [3] is incorrect. The error occurs in the proof of Theorem 8, and is due to my having interpreted certain group constructions as cartesian products, whereas they are in fact wreath products. The correct version of Theorem 8 is obtained by replacing cartesian products with wreath products in the appropriate places in Definition 5. The result, however, is easier to see than to state.

Professor Paul L. Sperry, when reviewing the paper [1] for Mathematical Reviews, has drawn the author's attention to an error in the formulation of Theorem 4.

The hypotheses on the abelian groups Ci, i ∈ I, in Theorem 4 were incorrectly stated. (They implied, for one thing, that the slender group A is divisible, an absurdity.) The correct formulation is as follows.