A product space X × Y is
rectangularly normal if every continuous real-valued function defined on a closed
rectangle A × B in X × Y can be continuously extended onto X × Y . It is known
that products of normal spaces with locally compact metric spaces are rectangularly
normal. In this paper we prove the converse of this theorem by showing there exists a
normal space X such that its product X ×M with a metric space M is rectangularly
normal if and only if M is locally compact, thus answering positively a question
raised by E. Michael.
Other related results are obtained; in particular, we show there exists
a normal space X and a countable metric space M with one non-isolated
point such that the product space X × M is not rectangular (in the sense of
Pasynkov).
