TITLE: A space in which sequences have unique limits but compact sets need not be closed
QUESTION [10 upvotes]: A topological space is KC if every compact subspace is closed.
A topological space is US if every convergent sequences has exactly one limit.
Does someone know an easy example of a US space which is not KC?
Thanks.

REPLY [5 votes]: I refer to COROLLARY 1 of This Article.
In COROLLARY 1 of it, $X^+$ denotes the one point compactification of the topological space $X$:

COROLLARY: Let $X$ be a Hausdorff space.Then:
(a) $X^+$ is always $US$.
(b) $X^+$ is $KC$ if and only if $X$ is a $\kappa$ space.

So it suffices to choose a Hausdorff space $X$, which is not $\kappa$ space. then $X^+$ is $US$ but is not $KC$.
PS: The topological space $X$ is called $\kappa$ space, if A subspace $A$ is closed in $X$ if and only if $A \cap K$ is closed in $K$, for all compact subset $K \subset X$.