Abstract:Let $L$ be an Archimedean Riesz space with a weak order unit $u$. A sufficient condition under which Dedekind [$\sigma $-]completeness of the principal ideal $A_{u}$ can be lifted to $L$ is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of $C(X)$-spaces. Similar results are obtained for the Riesz spaces $B_{n}(T)$, $n=1, 2,...$, of all functions of the $n$th Baire class on a metric space $T$.
Keywords: Dedekind completeness, spaces of continuous functions, spaces of Baire functions
AMS Subject Classification: 46A40, 26A99, 46B30