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Fine topology on function spaces. (English) Zbl 0614.54014
The author studies certain fine topologies that can be imposed on the space C(X,Y) of all continuous mappings from a Tikhonov space X into a uniform Tikhonov space Y. From the observation that the uniformity on Y induces a uniformity on C(X,Y) he proceeds to derive a number of results of which the following will give some idea of their scope. If X is compact, then all compatible uniformities on Y will generate the same topology on C(X,Y), namely, the compact-open topology. If Y is completely metrizable, then C(X,Y) with the fine uniform topology induced by the fine covering uniformity on Y will be a Baire space. In case Y is the real line with metric \(\rho\leq 1\), the author writes C(X) for C(X,Y) and denotes by \(C_{f\rho}(X)\) the topology generated under translation by neighborhoods of the zero-function \(f_ 0\) having the form \(\{\) \(g\in C(X):\rho (g(x),0)<\phi (x)\) for all \(x\in X\}\) as \(\phi\) runs over the continuous mappings from X into the positive real numbers. If X is normal and \(f_ 0\) has a countable base in \(C_{f\phi}(X)\), then X must be countably compact. Finally, if C(X) is first countable in the fine uniform topology, then X must be pseudocompact.
Reviewer: J.V.Whittaker

MSC:
54C35 Function spaces in general topology
54E15 Uniform structures and generalizations
54D65 Separability of topological spaces
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