We show that if X is a complex hereditarily indecomposable space, then every operator from a subspace Y of X to X is of the form λI+S, where I is the inclusion map and S is strictly singular.

We prove that it is consistent with the axioms of ZFC, relative to the consistency of ZFC itself, that the Baire number of the Stone representation space of the Boolean algebra [Pscr ](ω)/fin equals the distributivity number of [Pscr ](ω)/fin. This answers a question of Balcar, Pelant and Simon in [1].

The purpose of this note is to obtain a restriction on the fundamental groups of non-elliptic compact complex surfaces of class VII in Kodaira's classification [9]. We recall that these are the compact complex surfaces with first Betti number one and no non-constant meromorphic functions. This seems to be the class of compact complex surfaces whose structure is least understood. The first and simplest examples are the general Hopf surfaces [9, III]. Then there are various classes of examples found by Inoue [5, 6], and which have been studied in more detail in [11]. The only known topological restriction beyond the first Betti number is that intersection form in two-dimensional homology is negative definite. There seems to be little known as to how wide this class of surfaces is. We prove the following theorem.

Let S be a locally compact Hausdorff space, and let C0(S) be the Banach space of complex-valued continuous functions on S vanishing at infinity. It is known [1, p. 93] that every isometry U on C0(S) has the form

(Uf)(s)=θ(s)f(γ(s)) (s∈S), ([midast ])

where γ is an automorphism of S, and θ is a continuous function on S satisfying [mid ]θ(s)[mid ]=1 for all s∈S. Recall that a norm on a Banach space X is maximal if there is no equivalent norm for which the group of isometries is strictly larger. It is easy to see that the norm of C0(S) is not maximal if S contains a finite number of isolated points (S has at least two isolated points) [3, Remarks on Theorem 8.2]. In [3, 5, 6], Kalton and Wood proved the following theorem.

It is shown that for every compact group G, L1(G)ˆ is unique and minimal among all the closed subsets I of M(G)** such that I is a proper (≠0, ≠M(G)**) algebraic ideal, and such that I is solid with respect to absolute continuity; that is, n∈L1(G)ˆ whenever n∈M(G)** and n[Lt ]μ∈L1(G)ˆ.

Le théorème fondamental de la théorie locale des surfaces affirme que toute immersion C3 d'une surface dans R3 est déterminée, è congruence près, par sa première et deuxième forme fondamentale. Nous montrerons dans cette note qu'une surface complète C5 de R3 avec [mid ]dH[mid ]≠0 est en fait essentiellement détermineé, à congruence près, par la première forme fondamentale et la trace de la seconde, i.e. par la métrique et la fonction courbure moyenne H.

Let X be a space that has the homotopy type of a finite simply connected CW complex. We denote by [Escr ](X) the group of homotopy classes of self-homotopy equivalences of X. This group has been extensively studied (see [1] for a survey). In this paper we consider the subgroup [Escr ]n#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on the homotopy groups πi(X) for i[les ]n, and the subgroup [Escr ]#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on all the homotopy groups. Our first result is as follows.

The notion of an asymptotic cycle is extended to flows on non-compact spaces, and it is proved that for a suitable class of flows on manifolds it is true that the set of points with asymptotic cycles defined has measure one with respect with every invariant probability measure.

An explanation of a duplication identity in law involving the variances of the Brownian bridge and Brownian motion is given, with the help of an elementary transformation in time and space of a two-dimensional Brownian motion.

Let p be a point of a Lorentzian manifold M. We show that if M is spacelike Osserman at p, then M has constant sectional curvature at p; similarly, if M is timelike Osserman at p, then M has constant sectional curvature at p. The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [3]; we use a different approach to this case.

Frank Adams was born in Woolwich on 5 November 1930. His home was in New Eltham, about ten miles east of the centre of London. Both his parents were graduates of King's College, London, which was where they had met. They had one other child — Frank's younger brother Michael — who rose to the rank of Air Vice-Marshal in the Royal Air Force.

In his creative gifts and practical sense, Frank took after his father, a civil engineer, who worked for the government on road building in peace-time and airfield construction in war-time. In his exceptional capacity for hard work, Frank took after his mother, who was a biologist active in the educational field.

1. Ordered spaces

Risking semantic incoherence, we propose in this paper to use the term ordered space to mean any totally ordered set X equipped with a topology [Tscr ], while retaining the term linearly ordered space for its usual role of denoting the special case in which [Tscr ] is the order topology: the topology generated by the order-open initial and final segments of X.

The categories that we shall consider are full subcategories of the category OrdTop of ordered spaces and order-preserving continuous maps. It is easy to see that — in contrast to the situation in the full subcategory LinOrdTop of linearly ordered spaces — every inverse spectrum in OrdTop has a limit and that OrdTop admits subspaces and quotient spaces. These limits, subspaces and quotient spaces are all constructed precisely as in the category Top and are given their naturally induced orderings: where, however, we note that, for the ordered space X to induce an ordering on a quotient space, the quotient space (construed as a decomposition) must partition X into intervals rather than into arbitrary subsets. (In the terminology of [1], these constructions define subspaces and quotient spaces in OrdTop as, respectively, the domains and codomains of extremal monomorphisms and epimorphisms.) The purpose of the paper is to identify the closures of LinOrdTop under these constructions.