Topologies on \(X\) as points within \(2^{{\mathcal P}(X)}\) (Q439300)

In the 1930's, \textit{G. Birkhoff} introduced and studied the order-theoretic aspects of the set \(Top(X)\) of all topologies on a set \(X\), [Fundam. Math. 26, 156--166 (1936; Zbl 0014.28002)]. The present work analyzes families of subsets of a set \(X\) naturally identified with members of the Boolean space \(2^{P(X)}\) (a product of the discrete space \(2 = \{0,1\}\)): The main interest lies on the set \(Top(X)\) and on the set \(Lat(X)\) (respect., \(LatB(X)\)) of all sublattices (respect., \textit{bounded} sublattices) of \(P(X)\). Many sets are shown to be closed subsets: For instance, the set of all \(T_1\) topologies on \(X\) is closed in \(Top(X)\) and the sets \(Lat(X)\) and \(LatB(X)\) are closed in \(2^{P(X)}\).NEWLINENEWLINEThe main results in the paper are: (i) the subset \(Top(X) \subset LatB(X)\) is a dense and co-dense subset with empty interior; (ii) \(LatB(X)\) is a Hausdorff compactification of \(Top(X)\); (iii) \(Top(X)\) is not locally compact. Also the authors introduce an expansion of the first-order language of Boolean algebras by a unary predicate symbol. In this language it is proved that a subspace of \(2^{P(X)}\) is compact when it has a definition expressible by a collection of universal sentences, thus \(Top(X)\) is not first-order definable by first-order universal sentences.