Abstract
F denotes a nontrivially non-Archimedean valued field with rank one, X an ultraregular space and $C(X,F,p)$ is the vector space $C(X,F)$ of all continuous functions from X into F with the topology p of pointwise convergence. We show that $C(X,F,p)$ is a bornological space if and only if X is a Z-replete space. Also, some results are found concerning the compact-open topology c and we make a comparison with that case as studied by Bachman, Beckenstein, Narici and Warner.
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