A topological space $(X,\tau)$ is called $C-C$ iff the closed sets in $X$ coincide with the compact sets in $X$.
A topological space is called a $US$-space provided that each convergent sequence has a unique limit.
We know that a topological space $C-C$ is compact and $T_1$, because singleton sets are compact.
(1): Is there an example of a compact $T_1$ space that is not $C-C$ ?
(2): Is there an example of a compact $US$ space that is not $C-C$ ?
Let $X$ be any infinite set, and let $\tau$ be the cofinite topology on $X$. Then $\langle X,\tau\rangle$ is compact and $T_1$, and every subset of $X$ is compact, but the only closed sets in $X$ are the finite sets and $X$ itself, so $X$ is not $C$-$C$.
Let $X=\left\{\frac1n:n\in\Bbb Z^+\right\}\cup\{0,p\}$, where $p$ is a non-principal ultrafilter on $\Bbb Z^+$. Points of $X\setminus\{p\}$ have their usual nbhds. For each $A\in p$ let $B_A=\{p\}\cup\left\{\frac1n:n\in A\right\}$, and take $\{B_A:A\in p\}$ as a local base at $p$. $X$ is compact, because every nbhd of $0$ contains all but finitely many points of $X$. If $\sigma=\langle x_n:n\in\omega\rangle$ is a convergent sequence in $X$, then either $\sigma$ is eventually constant, in which case it has a unique limit, or it converges to $0$, so $X$ is $US$. Finally, $X\setminus\{p\}$ is a compact subset of $X$ that is not closed in $X$.