Dispersed images of topological spaces and uncomplemented subspaces of $C(X)$

Abstract:

Let $\Gamma (\alpha )$ denote the set of ordinals not exceeding $\alpha$ with its interval topology. We show that, if $X$ is a $0$-dimensional Hausdorff space and $\alpha$ is a denumerable ordinal such that the $\alpha$th derived set of $X$ contains $n$ points where $n < \omega$, there is a map of $X$ onto $\Gamma ({\omega ^\alpha } \cdot n)$. Maps of completely regular spaces into the unit interval are considered and a noncompact analogue of a theorem of Pełczyński and Semadeni is obtained. Finally, these results are used to give a simple proof to the following theorem: If $X$ is completely regular and ${X^{(\omega )}} \ne \emptyset$, there is an uncomplemented subspace $H$ of $C(X)$ which is isometrically isomorphic to $C(Y)$ for some compact metric space $Y$.