**
Lajos Soukup**

Nagata conjectured that every -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 of size
such that in
Nagata's conjecture holds
for each first countable regular space from the ground model (i.e.
if a first countable regular space is an -space in then
it is homeomorphic to a closed subspace of
the product of a countably compact space and a metric space in ).
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 .
As a corollary, we also obtain that
every first countable regular space from the ground model
has a maximal first countable extension in model .

**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