WHICH TOPOLOGIES HAVE IMMEDIATE PREDECESSORS IN THE POSET OF HAUSDORFF TOPOLOGIES?

We study which topology have an immediate predecessor in the poset of Sigma(2) of Hausdorff topologies on set X. We show that certain classes of H-closed topologies, do have predecessors. and we give examples of second countable H-closed topologies which are not upper Sigma(2.)

NATURAL TOPOLOGIES ON COLOMBEAU ALGEBRAS

We define intrinsic, natural and metrizable topologies T(Omega), T, T(s,Omega) and T(s) in G(Omega), (K) over bar, G(s)(Omega) and (K) over bar (s) respectively. The topology T(Omega) induces T, T(s,Omega) and T(s). The topologies T(s,Omega) and T(s) coincide with the Scarpalezos sharp topologies.