Abstract
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 X F =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 X F 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 C p(X F ) of real-valued continuous functions on X F with the pointwise topology is hereditarily Baire.
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