Some properties of open ultrafilters

We study the space of ultrafilters for an arbitrary topological space under natural equipment similar to that used in construction of Stone compactum. It is showed that above ultrafilters space is extremely disconnected compactum. We consider the families of sets in the space of ultrafilters majorizing (any time) the filter of open neighborhoods of a fixed point of initial space. Conditions guaranteeing mutually disjointness and distinguishability of sets for the given family are investigated; in particular, a special axiom of separability connected with support of mentioned distinguishability is introduced.