Here is an answer to Dirk's last question, ``Is"Is there a class of non_Hausdorff spaces in which convergent sequences have unique limits''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', "Between $\mathrm T_1$ and $\mathrm T_2$" (MSN), subsumes, references, or implies all of the following).)
In a KC-space, convergent sequences have unique limits.
(Suppose xn-->x$x_n\to x$ in the KC space X$X$. The set {x,x1,x2,..}$\{x,x_1,x_2,\dotsc\}$ is compact and hence closed. Thus, if y$y$ is not in the set {x,x1,x2$\{x,x_1,x_2,\dotsc\}$,..} then the open set X minus {x,x1,x2,..}$X \setminus \{x,x_1,x_2,\dotsc\}$ shows it is false that xn-->y$x_n\to y$. Thus, if xn-->y$x_n\to y$, then y=x$y=x$ or y=xn$y=x_n$ for some n$n$. If y=xn$y=x_n$ for infinitely many indices n$n$ then y=x$y=x$ (since every KC space is T1 $\mathrm T_1$ (since singletons are compact) and since constant sequences have unique limits in a T1$\mathrm T_1$ space). If y=xn$y=x_n$ for finitely many indices $n$ then (deleting y$y$ from the sequence x1,x,2...$x_1,x_2,\dotsc$) we are left with a subsequence zn-->x$z_n\to x$, the knowledge that y$y$ is not znequal to any $z_n$, and the knowledge that y$y$ is in the set {x,z1,z2$\{x,z_1,z_2,\dotsc\}$,...} and we conclude y=xthat $y=x$).
To exhibit a large class of non-Hausdorff KC spaces, let X$X$ be a non-locally-compact metric space ( forfor example, the rationals) and let Y=X U {y}$Y=X \cup \{y\}$ denote the Alexandroff compactification of X$X$ ( ii.e. V, $V$ is open in Y$Y$ if V$V$ is open in X$X$ or if Y\V$Y\setminus V$ is a compact subspace of X$X$).
The space Y$Y$ is a KC space, but Y$Y$ is not Hausdorff.
