A perfectly normal space which is not realcompact.
by
Fernando Hernández-Hernández
York University
Coauthors: Tetsuya Ishiu (University of California, Irvine)

In 1976 R.L. Blair asked: Is there, in ZFC, a perfectly normal space which is not realcompact?

Two examples were already known: the Ostaszewski line from \diamondsuit and the discrete space of that measurable cardinality. Blair conjectured further that MA+\lnotCH implies that every perfectly normal space of cardinality less than the first measurable is realcompact . We settle this conjecture by proving

Theorem. It is consisting with MA+\lnot CH that there is a perfectly normal nonrealcompact space of cardinality \aleph₁.

The example is obtained by refining the order topology on \omega₁ in a forcing extension.

Date received: February 9, 2003