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
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
). 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
.