Critical ideals for compact spaces

Abstract

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