Critical ideals for compact spaces

For each countable ordinal $\alpha$, we introduce an ideal $con{v}_{\alpha }$ and use it to characterize the class of all compact countable spaces which are homeomorphic to the space ${\omega }^{\alpha }\cdot n+1$ with the order topology. The characterization is expressed in terms of finding a convergent subsequence defined on a set not belonging to $con{v}_{\alpha }$.