Filters, consonance and hereditary Baireness

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 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. © 2000 Elsevier Science B.V. All rights reserved.