Let X be the set of natural numbers. I would to prove that there is a unique proper minimal dense subset of $\beta(N)$ which is the Stone-Čech compactification. so by the definition we know that N is a dense. I have proved it is minimal but how can I prove it is unique. I mean if I assume there is another dense subset of $\beta(N)$, how can I prove it is equal to N.
Also I would to prove that $\beta(N)-N$ has no minimal dense subset. I assumed it has a minimal dense but I have no clue how to get the contradiction.
Any dense subset $D$ of $\beta \Bbb N$ must contain all $n \in \Bbb N$, as $D$ must intersect the non-empty open set $\{n\}$. So $\Bbb N \subseteq D$ and so $\Bbb N$ is the minimal dense subset of $\beta \Bbb N$.
As to $\Bbb N^\ast=\beta \Bbb N \setminus \Bbb N$: This space has no isolated points and is $T_1$ which implies that for any dense subset $D$ of it, and any $d \in D$, $D\setminus \{d\}$ is also dense, so there never can be a minimal (wrt to inclusion) dense subset of the space.