Compact zero-dimensional $T_2$-topologies on $\mathbb{N}$

Question

Let $\tau$ be a compact topology on $\mathbb{N}$ such that for every two points $m\neq n\in \mathbb{N}$ there is a clopen set $U$ containing $m$ but not $n$.

Is $(\mathbb{N},\tau)$ isomorphic to $\omega+1$ with the interval topology?

Answer 1

It seems the following.

Of course not necessarily. There is a lot even Hausodorff countable compact spaces. By Sierpinski-Mazurkiewicz Theorem, each of them is homeomorphic to $\omega^\alpha\cdot n+1$.

Answer 2

Not necessarily: it could be homeomorphic to a disjoint union of any finite number of copies of $\omega+1$, for instance. In fact it can be homeomorphic to any infinite countable successor ordinal: they're all compact, zero-dimensional Hausdorff spaces.