Nagata's conjecture and countably compactifications in generic extensions

Lajos Soukup

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 $V^P$). By a result of Morita, it is enough to show 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 $V^P$.

Key words and phrases: countably compact, compactification, countably compactification, countably compactifiable, first countable, maximal first countable extension, $M$-space, forcing, Martin's Axiom

2000 Mathematics Subject Classification: Primary: 54D35, Secondary: 54E18, 54A35, 03E35

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