Realcompactifications of products of ordered spaces

Abstract

The equality υ ( X × Y ) = υ X × υ Y \upsilon (X \times Y) = \upsilon X \times \upsilon Y is studied for the case when one of the factors is a linearly ordered topological space (LOTS). Among the results obtained are the following: 1. If X X is any separable realcompact space and Y Y is any LOTS of nonmeasurable cardinal, then υ ( X × Y ) = υ X × υ Y \upsilon (X \times Y) = \upsilon X \times \upsilon Y . 2. If X X is a nonparacompact LOTS, then there is a paracompact LOTS Y Y such that υ ( X × Y ) ≠ υ X × υ Y \upsilon (X \times Y) \ne \upsilon X \times \upsilon Y . 3. For any pair X , Y X,Y of well-ordered spaces, υ ( X × Y ) = υ X × υ Y \upsilon (X \times Y) = \upsilon X \times \upsilon Y .