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Topologies on the space of continuous functions. (English) Zbl 0940.54023
The authors study the coincidence of the fine topology $$\tau_{\omega}$$ (Whitney topology, Morse topology, $$m$$-topology) on the set $$C(X,Y)$$ of all continuous functions from a topological to a metric space with some other function space topologies. (Note that the fine topology depends on the concrete metric on $$Y$$, not only on the underlying topology, whereas some of the compared topologies are completely topologically defined.) They show that $$\tau_{\omega}$$ coincides with the graph topology, if $$X$$ is countably paracompact and normal, with the Krikorian topology, if in addition the metric space $$Y$$ is separable (moreover, it is shown, that the Krikorian topology coincides with the graph topology, if $$Y$$ is a topological space with countable base and $$X$$ is countably paracompact and normal), with the topology of uniform convergence and the open-cover topology, if $$X$$ is countably compact and $$Y$$ paracompact and first countable. Using cardinal functions, some interesting lemmas concerning the fine topology are established. The equivalence of first countability, metrizability, complete metrizability, Čech-completeness and pseudocompactness is shown for $$(C(X,IR),\tau_{\omega})$$ without any requirements on $$X$$. Furthermore, the space $$W$$ of all ordinals less than the first uncountable ordinal is examined and the coincidence of uniform convergence, Krikorian, graph and open-cover topology on $$C(W,W)$$ is shown.

##### MSC:
 54C35 Function spaces in general topology 54E35 Metric spaces, metrizability 54E05 Proximity structures and generalizations 54C05 Continuous maps 54E15 Uniform structures and generalizations
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