Here is an answer to Dirk's last question, ``Is there a class of non_Hausdorff spaces in which convergent sequences have unique limits''?

Yes. The so called KC-spaces or maximal compact spaces. These are spaces such that every compact subspace is closed.

(The 1967 Monthly article of Wilansky `Between T1 and T2' subsumes, references, or implies all of the following).

In a KC-space, convergent sequences have unique limits.

(Suppose xn-->x in the KC space X. The set {x,x1,x2,..} is compact and hence closed. Thus if y is not in the set {x,x1,x2,..} then the open set X minus {x,x1,x2,..} shows it is false that xn-->y. Thus if xn-->y then y=x or y=xn for some n. If y=xn for infinitely many indices n then y=x (since every KC space is T1 (since singletons are compact) and since constant sequences have unique limits in a T1 space). If y=xn for finitely many indices then (deleting y from the sequence x1,x,2...) we are left with a subsequence zn-->x, the knowledge that y is not zn, and the knowledge that y is in the set {x,z1,z2,...} and we conclude y=x).

To exhibit a large class of non-Hausdorff KC spaces let X be a non-locally-compact metric space ( for example the rationals) and let Y=X U {y} denote the Alexandroff compactification of X ( i.e. V is open in Y if V is open in X or if Y\V is a compact subspace of X).

The space Y is a KC space but Y is not Hausdorff.