Nagata's conjecture and countably compact hulls in generic extensions

Abstract

Nagata conjectured that every M -space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. Although this conjecture was refuted by Burke and van Douwen, and A. Kato, independently, but we can show that there is a c.c.c. poset P of size 2 ω such that in V P Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space X ∈ V is an M -space in V P then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in V P ). By a result of Morita, it is enough to show that every first countable regular space from the ground model has a first countable countably compact extension in V P . As a corollary, we also obtain that every first countable regular space from the ground model has a maximal first countable extension in model V P .