# Article

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Keywords:
Dedekind completeness; spaces of continuous functions; spaces of Baire functions
Summary:
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 $B_{n}(T)$, $n=1, 2, \dots$, of all functions of the $n$th Baire class on a metric space $T$.
References:
[1] Kuratowski K.: Introduction to Set Theory and Topology. Polish Scientific Publishers, Warszawa, 1997. MR 0346724 | Zbl 1081.54501
[2] Kuratowski K., Mostowski A.: Set Theory. Polish Scientific Publishers, Warszawa, 1996. MR 0485384 | Zbl 0337.02034
[3] Luxemburg W.A.J., Zaanen A.C.: Riesz Spaces I. North-Holland, Amsterdam, 1971.

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