Filters, consonance and hereditary Baireness
Résumé
A topological space $X$ 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 $\cal F$ on the set of natural numbers N, let $N_F=N\cup\{\infty \}$$ be the space for which all points in N are isolated and the neighborhood system of ∞ is $\{A\cup\{\infty\}: A\in {\cal F}\}$. We give a combinatorial characterization of the class Φ of all filters $\cal F$ such that the space $N_F$ is consonant and all its compact subsets are finite. It is also shown that a filter $\cal F$ belongs to Φ if and only if the space $C_p(N_F)$ of real-valued continuous functions on $N_F$ with the pointwise topology is hereditarily Baire.