Almost maximal spaces
A topological space \(X\) is called almost maximal if it is without isolated points and for every \(x \in X\) there are only finitely many ultrafilters on \(X\) converging to \(x\). We associate with every countable regular homogeneous almost maximal space \(X\) a finite semigroup \(ULT(X)\) so that if \(X\) and \(Y\) are homeomorphic, \(Ult(X)\) and \(Ult(Y)\) are isomorphic. Semigroups \(Ult(X)\) are projectives in the category \({F}\) of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under Martin's axiom, for each projective \(S\) in \({F}\), there is a countable almost maximal topological group \(G\) with \(Ult(G)\) isomorphic to \(S\). The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces \(X\) with \(Ult(G)\) being a chain of idempotents. Finally a number of interesting open questions are given at the end of the paper.
scientific article; zbMATH DE number 5123617