Abstract
F denotes a nontrivially non-Archimedean valued field with rank one, X an ultraregular space and C ( X , F , p ) C(X,F,p) is the vector space C ( X , F ) 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 ) 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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