Nagata’s Conjecture and Countably Compactifications in Generic Extensions

Nagata conjectured that everyM -space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. This conjecture was refuted by Burke and van Douwen, and A. Kato, independently. However, we can show that there is a c.c.c. poset P of size 2 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 ). In fact, we show that every first countable regular space from the ground model has a first countable countably compact extension in V P , and then apply some results of Morita. As a corollary, we obtain that every first countable regular space from the ground model has a maximal first countable extension in model V P .