A Surprising Similarity Between a Closed n‐cell and βRn
David P. Bellamy, Beverly Diamond · Annals of the New York Academy of Sciences · 1993
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