Filters, consonance and hereditary Baireness

A topological space is called consonant if, on the set of all closed subsets of X, the co-compact topology coincides with the upper Kuratowski topology. For a filter F on the set of natural numbers ω, let XF=ω∪{∞} be the space for which all points in ω are isolated and the neighborhood system of ∞ is {A∪{∞}: A∈F}. We give a combinatorial characterization of the class Φ of all filters F such that the space XF is consonant and all its compact subsets are finite. It is also shown that a filter F belongs to Φ if and only if the space Cp(XF) of real-valued continuous functions on XF with the pointwise topology is hereditarily Baire.