Irreducible filters and sober spaces

A filterF on a convergence space is called irreducible, iff the set convF of convergence points ofF belongs toF. A space is sober, iff for every irreducible filterF there is a unique point x with convF=conv x. The categorySob-Conv of sober convergence spaces is a full productive, but not a reflective subcategory of the categoryConv of convergence spaces and continuous maps. For a topological space (X,t) the following are equivalent: (i) for every irreducible filterF on (X,t) there is a point x withF=x (ii) (X,t) is both sober and T_D (iii) every subspace of (X,t) is sober (iv) every topological space finer than (X,t) is sober (v) whenever (Y,s) is a T_o-space whose latticeO(Y,s) of open sets is isomorphic toO(x,t), then (Y,s)(X,t). The categorySob-T ₁ß of sober T₁-spaces is the greatest epi-reflective subcategory ofTop consisting of sober spaces, moreoverSob-T ₁ß is a dis-connectedness in the sense of Preuß- Arhangelskii-Wiegandt (generated by all irreducible spaces), henceSob-T ₁ is (extremal epi)-reflective inTop. It is strictly between T₁ and T₂ and different from various sorts of weak Hausdorffness discussed in the literature.