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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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