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

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.

### MSC:

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

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