Nagata's conjecture and countably compactifications 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. 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 $2^{\omega}$ 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 $X\in V$ 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 $V^P$.