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Keywords:
Dugundji extension theorem; non-archimedean space; space of continuous functions; 0-dimensional space
Dugundji extension theorem; non-archimedean space; space of continuous functions; 0-dimensional space
Summary:
We prove a non-archimedean Dugundji extension theorem for the spaces $C^{\ast }(X,\mathbb {K})$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $\mathbb {K}$. Assuming that $\mathbb {K}$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T\colon C^{\ast }(Y,\mathbb {K})\rightarrow C^{\ast }(X,\mathbb {K})$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $\mathbb {K}$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ of an ultraregular space $X$ is a retract of $X$.
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