A local property associated with the semilattice of compactifications of a space (Q1262570)

If K(X) is the set of all compactifications of a Tychonoff space X and \(Y,Z\in K(X)\), then we say that \(Y\leq Z\) if there exists a continuous function from Z onto Y which is the identity on X. The paper under review is devoted to the problem of when the complete upper semilattice K(X) is a lattice. Following the approach of \textit{Y. Ünlü} [General Topol. Appl. 9, 41-57 (1978; Zbl 0389.54015)] the author considers the set D(X) of all upper semicontinuous decompositions of X partially ordered by refinement. If Y is a compact space and \(X\subset Y\) then \(D_ Y(X)\) denotes the set of all \(P\in D(Y)\) such that \(\{\) \(y\}\in P\) for every \(y\in Y-X\). The complete lower semilattice \(D_ Y(X)\) is called to be locally a lattice at a point \(y\in Y-X\) if there exists a closed neighbourhood E of y in Y such that \(D_ E(E\cap X)\) is a lattice. The main result says that \(D_ Y(X)\) is a lattice iff it is locally a lattice at every point of Y-X. In terms of the theory of compactifications it means that K(X) is a lattice iff it is locally a lattice at every point of X. In particular, if \(\beta\) X-X contains a sequence converging to a point \(x\in X\), then K(X) is not locally a lattice at x. Hence, K(X) is not a lattice in this case. This gives the result of \textit{J. Flachsmeyer} and \textit{J. Visliseni} [Dokl. Akad. Nauk SSSR 165, 258-260 (1965; Zbl 0142.209)]. The main result is also used to show that if \(U\subset X\) is open and K(X) is a lattice, then K(U) is a lattice as well and if X is the disjoint union of a family \(\{X_ i:i\in I\}\), then K(X) is a lattice iff \(K(X_ i)\) is a lattice for all \(i\in I\).