TITLE: A non-Hausdorff space with unique limit
QUESTION [5 upvotes]: Can I find a topological space $X$ such that every convergent sequence in $X$ has a unique limit in $X$, but $X$ is not Hausdorff?

REPLY [11 votes]: The co-countable topology on an uncountable set $X$ (like the reals) (the only closed sets are $X$ and all sets that are at most countable) is such a space.
The only convergent sequences are eventually constant and so the limits of convergent sequences are unique. But all non-empty open sets intersect, so $X$ is in fact anti-Hausdorff.