Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x ∈ 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 MA, 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 ( X ) being a chain of idempotents.