Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 157-164, 2013

# A non-archimedean Dugundji extension theorem

## Jerzy KĄkol, Albert Kubzdela, Wieslaw Śliwa

Jerzy KĄkol, Faculty of Mathematics and Informatics, A. Mickiewicz University, Poznań, 61-614 Poland, e-mail: kakol@amu.edu.pl; Albert Kubzdela, Institute of Civil Engineering, Poznań University of Technology, Poznań, 61-138 Poland, e-mail: albert.kubzdela@put.poznan.pl; Wieslaw Śliwa, Faculty of Mathematics and Informatics, A. Mickiewicz University, Poznań, 61-614 Poland, e-mail: sliwa@amu.edu.pl

Abstract: 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 $T C^{\ast}(Y,\mathbb{K})\rightarrow C^{\ast}(X,\mathbb{K})$ if $X$ is collectionwise normal or $Y$ is Lindelof 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 $X$.

Keywords: Dugundji extension theorem, non-archimedean space, space of continuous functions, 0-dimensional space

Classification (MSC 2010): 46S10, 54C35

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