Finite-point order compactifications

Abstract

After the characterization of 1-point topological compactifications by Alexandroff in 1924, n-point topological compactifications by Magill [4] in 1965, and 1-point order compactifications by McCallion [5] in 1971, spaces that admit an n -point order compactification are characterized in Section 2. If X* and X** are finite-point order compactifications of X, sup{X*, X**} is given explicitly in terms of X* and X** in § 3. In § 4 it is shown that if an ordered topological space X has an m-point and an n-point order compactification, then X has a k-point order compactification for each integer k between m and n. The author is indebted to Professor Darrell C. Kent, who provided assistance and encouragement during the preparation of this paper.