C-scattered type and minimal compactifications with countable remainder

For each α<ω1, we define a C-scattered countable metric space Nα of level α which is contained as a closed subset in every metric C-scattered space of level α. We characterize the spaces Nα topologically and prove that they are minimal spaces. We use the results to refine results of Zippin and Hoshina by showing that a rim-compact separable metrizable space X with Rω0(X)=∅ has a minimal metrizable compactification with remainder homeomorphic to Nα or the direct sum of countably many copies of Nα, for a certain α≤ω0.