Hausdorff compactifications and zero-one measures (Q2758160)

In [Am. J. Math. 86, 602-607 (1964; Zbl 0129.38101)] \textit{O. Frink} introduced the concept of Wallman-type compactification and posed the question whether each Hausdorff compactification of a Tikhonov space \(X\) is a Wallmann-type compactification. In [Sov. Math., Dokl. 18, 567-571 (1977); translation from Dokl. Akad. Nauk SSSR 233, 1056-1059 (1977; Zbl 0371.54048)] \textit{V. M. Ulyanov} obtained a negative answer to Frink's question: each Wallman-type compactification of \(X\) can be constructed via the space \(\max (\mathcal B)\) of maximal ideals of a normal Wallman base \(\mathcal B\) of \(X\) and not every Hausdorff compactification of \(X\) can be obtained in this way. NEWLINENEWLINENEWLINEThe authors provide a positive answer to the following question: is it possible to correlate (in a canonical way) to each Tikhonov space \(X\) a Boolean algebra \(B_X\) and a set \(\mathcal L_X\) of sublattices of \(B_X\) in order to obtain that the set of all, up to equivalence, Hausdorff compactifications of \(X\) is represented by the set \(\{\max (L); L\in \mathcal L_X\}\)? This leads to the solution of a parallel problem: it is known that each Wallman compactification can be constructed via the space of suitable measures on a Boolean algebra and the authors prove that the set of all, up to equivalence, Hausdorff compactifications of a Tikhonov space \(X\) can be represented by the set \(\{I_{ur}(L); L\in \mathcal L_X\}\), where \(I_{ur}(L)\) is a certain set of zero-one measures. NEWLINENEWLINENEWLINEThe paper under review consists of a well-written concise (hi)story of the famous Wallman-type compactification problem, an exposition of technical preliminaries [mostly the results of two previous papers by the same authors, Ann. Mat. Pura Appl., IV. Ser. 169, 87-108 (1995; Zbl 0881.54028); Math. Proc. Camb. Philos. Soc. 119, No. 2, 321-339 (1996; Zbl 0876.54020)], and the results. The two main results are in Theorem 3.17. Part (a) is the answer to the question stated above. Part (b) is the solution of the parallel problem.