Non-archimedean nuclearity and spaces of continuous functions. (English) Zbl 0746.54004

de Grande-de Kimpe, N.; Navarro, S.

Indag. Math., New Ser. 2, No. 2, 201-206 (1991).

Let \(X\) be a zero-dimensional topological Hausdorff space. Let \(K\) be a non-archimedean valued, complete field \(K\), non-trivially valued. \(C(X):=\{f: X\to K; f\text{ is continuous}\}\). Let \(\tau_ c\) be the compact-open topology on \(C(X)\), \(\tau_ p\) the topology of pointwise convergence and \(\tau_{\sigma}\) the weak topology. The authors prove:
The following are equivalent: (i) the space \((C(X),\tau_ c)\) is nuclear; (ii) every \(\tau_ c\)-bounded subset of \(C(X)\) is \(\tau_ c\)- compactoid; (iii) every compact subset of \(X\) is finite; (iv) on \(C(X)\) the topologies \(\tau_ c\) and \(\tau_ p\) coincide; (v) on \(C(X)\) the topologies \(\tau_ c\) and \(\tau_{\sigma}\) coincide; (vi) every \(\tau_ c\)-compactoid subset of \(C(X)\) is \(\tau_ c\)-compactoid; (vii) every \(\tau_ p\)-compactoid subset of \(C(X)\) is \(\tau_ c\)-compactoid; (viii) the space \((C(X),\tau_ c)\) is of countable type and in \(C(X)\) every \(\tau_ p\)-Cauchy sequence is \(\tau_ c\)-Cauchy.

Reviewer: M.Valdivia (Valencia)

MSC:

54C35	Function spaces in general topology
46A99	Topological linear spaces and related structures

Keywords:

non-archimedean nuclearity; spaces of continuous functions; compact-open topology; topology of pointwise convergence; weak topology

References:

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