We give a relatively simple proof of the theorem on the spectral sequence of a fibre-space in the sense of Kan. The method consists in the reduction to locally trivial fibre-spaces by minimal fibrings and in the use of a very simple acyclic model theory (without degeneracies).

In chapter I, E = {nk} ⊂ Z is shown to be a Sidon set if and only if (**). For each x ∈ T,

.

Let E ⊂ Z+ be a lacunary sequence. In chapter II, it is constructively shown that the characteristic function of E is uniformly approximable by Fourier–Stieltjes coefficients; i.e. ϕE ∈ M(T)∧−.

The spectrum of the hydrogen energy operator

(Δ is the Laplacian and r is the distance from the origin) consists of the non-negative real axis and a sequence of negative eigenvalues of finite multiplicities converging to O. In the present study we are interested in finding sufficient conditions on a potential q(x) such that the spectrum of the operator

in En has a ‘hydrogen-like’ spectrum, i.e. a spectrum consisting of

(a) the non-negative real axis,

(b) at most a denumerable set of negative eigenvalues of finite multiplicities having zero as its only possible limit point.

Given any triangulation of a topological n-cell, In, there is an integer k such that the kth barycentric subdivision of the triangulation is simplicially collapsible (3). In the following, we show that given any integer k, we can construct a triangulation of an n-cell, n > 2, whose kth barycentric subdivision is not simplicially collapsible.

A group G of homeomorphisms of a topological space X will be called discontinuous if

(1) the stabilizer of each point of X is finite, and

(2) each point x ∈ X; has a neighbourhood U such that any element of G not in the stabilizer of x maps U outside itself (i.e. if gx ≠ x then U ∩ gU is empty). The purpose of this note is to prove the following result.

1. In this paper, I apply the results of (4) to a class of function spaces distinguished by their invariance under a certain family of operators. This class contains the Lp spaces (indeed, all Orlicz spaces) and the symmetric Köthe spaces of (5). My original objective was simply a more general expression of the ideas of (5), but I found myself taking up a rather different approach.

In (2), Varopoulos examined the structure of topological groups that are the inductive limits of countably many locally compact Abelian groups. The purpose of this note is to show that the theory does not extend to the case of uncountably many groups. We give two examples, the first to show that the strict inductive limit of uncountably many compact Abelian groups need not be complete, the second to show it need not be separated by its continuous characters. The treatment of this latter half follows closely that given for topological linear spaces by Douady in (l).

Gel'fand and Šilov(1) have introduced the concept of compatibility of norms on a vector space, defining two norms to be compatible if each sequence which is Cauchy with respect to both norms and converges to zero with respect to one norm, also converges to zero with respect to the other. They have found it convenient to build up their exposition of spaces of generalized functions (2) by developing the theory of locally convex spaces whose topology can be defined by a countable set of pairwise compatible norms. The author (3) has examined the properties of locally convex spaces which have been topologized by making a set of linear mappings into locally convex spaces continuous, and has shown that if this method is applied to a set of closed operators on Hilbert space, then the resulting spaces have topologies generated by a set of pairwise compatible norms. The purpose of this paper is to generalize this result by putting it in its natural context as a theorem on uniformities which have been defined by making a closed set of mappings uniformly continuous. The closed mappings of the title are mappings with closed graph.

The purpose of this paper is to give a complete list of those nilpotent groups which can be the fundamental groups of connected, closed, compact but possibly non-orientable 3-manifolds. The starting point is the following theorem of Reidmeister, which is given a neat proof in (1).

An ‘inner measure’ analogous to Lebesgue inner measure and associated with a von Neumann algebra is constructed on the linear sets of a Hilbert space. We supposed to be of finite type and countably decomposable, as these restrictions will be necessary for some of the results obtained. As remarked by Dye in (2), Type II1 algebras have a structure analogous to that of a finite non-atomic measure space, the trace corresponding to the measure. We define a class of ‘measurable sets’ and obtain some of its properties. The development of the theory is indicated also from the starting point of a function-valued dimension function. The idea for the construction of such an inner measure comes from a remark in (3), paragraph 16·2, and the terminology of (3) and of (5) is used throughout.

Alexander's theorem (5) states that a topological space is compact if there is a sub-base, , for its closed sets such that every subclass of with the finite intersection property has a non-empty intersection. An analysis and extension of this is given here which has applications, inter alia, to problems concerning real-compactness (2).

In this paper we study partially ordered vector spaces X whose positive cone K possesses a base which defines a norm in X. A positive decomposition x = y − z of the element x is said to be minimal if ‖x‖ = ‖y‖ + ‖z‖. We proved in (6) that the property that every element of X has a unique minimal decomposition is equivalent to an intersection property for homothetic translates of the base. Section 2 of the present paper analyses this intersection property in much more detail and discusses possible generalizations of it.

In this paper I investigate the completed projective tensor product of two perfect Riesz spaces, and show how a natural order structure on this renders it also a perfect Riesz space. Sections 7–14 contain interesting order-topological properties of this tensor product. Finally, section 15 describes how the tensor product of function spaces may be represented as a function space, in the manner of (1 b).

Let G be a locally compact Hausdorff topological group, with µ the left Haar measure on G, and µ* the corresponding inner measure. If R denotes the real numbers, ℬ(G) denotes the Borel† subsets of G of finite measure, and VG = {µ(E):E∈ℬ(G)}, then, following Macbeath(4), we define ΦG: VG × VG→ R by

where AB denotes the product set of A and B in G.

A metric on a differentiable manifold induces a bilinear form on the tangent space at each point of the manifold. The set of tangent vectors orthogonal, with respect to this bilinear form, to the whole tangent space is a vector subspace of the tangent space: the metric is called non-degenerate or degenerate at the point according as the subspace is or is not empty. This paper is concerned with the geometry of manifolds having everywhere degenerate metrics.

MacLane(1) has introduced the class of functions with a dense set of asymptotic values on {|z| = 1} in the following sense:

Definition. Let f(z) be regular and not constant in the unit disc and let ξ be a point of {|z| = 1}. Suppose that there exists a curve Γξ in {|z| < 1} and a in the closed complex plane such that f(z) → a as z → ξ along Γξ. Then we say ξ ∈ Aa.

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the property

a partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

Some recent work by the authors (1) on the distribution of the residues of a cubic polynomial modulo an odd prime p led to the conjecture that, in general, two cubic polynomials with integer coefficients possessing the same residues modulo p (not necessarily occurring to the same multiplicity) are equivalent, that is are related by a linear transformation modulo p. The purpose of the present paper is to prove this conjecture. We establish the following theorem.

Let be a real algebra which is also a Banach space. Then is called a partially ordered Banach algebra if there is specified a non-empty subset of , called the positive cone, such that

and

(A 5) is 1-normal and closed. (We recall that is said to be 1-normal if

This paper continues the investigation begun in (6), and uses the definitions and notation of that paper.