Lattices of compactifications of Tychonoff spaces

Abstract

It is shown that, for a Tychonoff space X, the complete upper semilattice K(X) of compactifications of X is a lattice if either (1) βX⧹X is realcompact and C^∗-embedded in βX, or (2) βX⧹X is a P-space and cl_βX(βX⧹X) is an F-space. The concept of bounding lattice is introduced and examples of spaces X are given such that K(X) is a lattice but not a bounding lattice. A certain class of Tychonoff spaces X is constructed such that K(X) is a lattice.