When does the family of singular compactifications form a complete lattice? (Q1265146)

A singular compactification \(aX\) of \(X\) is one whose remainder \(aX\smallsetminus X\) is a retract of \(aX\). It is known that the infimum of any family of singular compactifications is again singular. A similar result for the supremum of singular compactifications does not hold. In this paper the author provides a method of recognizing those spaces \(X\) for which the supremum of all singular compactifications is \(\beta X\). In addition, he characterizes singular compactifications \(aX\) of \(X\) in terms of properties of \(C^*(X)\), the ring of bounded real-valued continuous functions on \(X\). He also proves that a space \(X\) for which \(\beta X\) is singular must be pseudocompact. (Reviewer's remark: the author is unaware of the paper [\textit{E. K. van Douwen}, General Topol. Appl. 9, 169-173 (1978; Zbl 0386.54008)], which contains the same result.).