(1): MAre all point of $\beta\omega$ isolated? why?(2): is it true that $\omega$ is open in $\beta\omega$ , so $\beta\omega - \omega$ is compact?

please give me more explain. why can we say the set $\omega$ corresponds to the set of principal ultrafilters, and the set $\beta\omega - \omega$ to the set of free ultrafiltrates?

(1): What you mean about" If we let $B_{n}=A_{n}\setminus(A_{0}\cup...\cup A_{n-1})$, then the sequence $(B_{n})_{n}$ is pairwise disjoint and $B_{n}\in\mathcal{V}_{m}$ iff $m=n$. If we let $B=\bigcup_{n}B_{2n}$, then $B\in\mathcal{V}_{m}$ if and only if $m$ is even." ?(2): What does it mean " clopen set"?

In topological KC - space , every compact set is closed.a topological space( X,τ ) is called minimal- KC if ( X,τ )is KC and there is no topology σ ⊂ τ such that ( X, σ ) is KC. so, X = βω is KC - minimal.is there any example except it that is minimal- KC but does not have non- trivial sequence?

what is your references? I have read Engelking, but it is not clear to me.

the first two lines are definitions of KC and US space.