David P. Bellamy

SHORT PATHS IN HOMOGENEOUS CONTINUA

Abstract Homogeneous arcwise connected metric continua are shown to, in effect, be arcwise connected by arcs of bounded length. Specifically, for any positive e, there is a natural number n such that every two points can be joined by an arc which is the union of n subarcs of diameter less than e.

Extreme points of some classes of analytic functions with positive real part and a prescribed set of coefficients

Let P denote the set of functions f(z) = i + a1z + a2z2 +… that are analytic in the unit disc and satisfy Re f (z)> 0 for |x|< 1. Let Bn={b1, b2,…bn:|bk|< 1, k=1,2,…, n) where n is a natural number, and let P(Bn)={f ∊ P : ak=2bk, k=1,…,n}. We prove that the set of extreme points of P(Bn) consists exactly of the functions of the form where .

Mapping Continua Onto the Cone Over the Cantor Set

Publisher Summary A compact metric continuum ( S ) can be mapped continuously onto the cone over the Cantor set if S contains an open set with uncountably many components. This chapter provides an alternative characterization with a help of a theorem stating that a compact metric continuum S can be mapped onto the cone over the Cantor set if it can be mapped onto the cone for every countable nonlimit ordinal α.

A Surprising Similarity Between a Closed n-cell and βRn

ABSTRACT. A compactification kRn of Rn is geometric if (a) the closure in kRn of every closed topological copy of Rn‐1 separates kRn, and (b) any two points in kRn can be separated by the closure of such a topological hyperplane. The first condition trivially holds if kRn is a perfect compactification. Preliminary relationships between these two properties are discussed. The closed n‐ball Bn with Sn‐1 as remainder and kRn are geometric compactifications, as is any perfect metrizable compactification of R2 with nondegenerate remainder. The property of being geometric seems to be rare among compactifications of Rn, and it is surprising that it is shared by two as different as Bn and kRn