Consonance and topological completeness in analytic spaces

Ahmed Bouziad (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

We give a set-valued criterion for a topological space $X$ to be consonant, i.e., the upper Kuratowski topology on the family of all closed subsets of $X$ coincides with the co-compact topology. This characterization of consonance is then used to show that the statement `every analytic metrizable consonant space is complete' is independent of the usual axioms of set theory. This answers a question by Nogura and Shakhmatov. It is also proved that continuous open surjections defined on a consonant space are compact covering.

Mots clés: Analytic space; consonant space; Polish space

Domaines: Topologie générale [math.GN]

Ahmed Bouziad. Consonance and topological completeness in analytic spaces. Proceedings of the American Mathematical Society, 1999, 127 (2), pp.3733--3737. ⟨hal-00373442⟩