On the lattice of one-point near-compactifications (Q1072146)

A space X is nearly compact if X is H-closed and Urysohn, i.e., \(X_ s\) is compact Hausdorff. Let Y be a Hausdorff space. A subset \(A\subseteq Y\) is an N-set if every cover of A by sets regular-open in Y has a finite subcover. A Hausdorff space is locally nearly compact if each point has an N-set neighborhood; equivalently, a space X is locally nearly compact iff \(X_ s\) is locally compact Hausdorff. The author shows that the lattice of one-point near-compactifications of a locally nearly compact space, which is not H-closed, has a projective maximum. The author has communicated to the reviewer that 3.1(i) is false.