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Properties related to the first countability in the fine and further topologies. (English) Zbl 1072.54012
The fine topology is defined on the set \({\mathbf C}(X,Y)\) of continuous functions from a topological space \(X\) into a metric space \((Y,d)\), and has applications in approximation theory, differential geometry, and rings of continuous functions. In this paper the author gives characterizations of first countability, sequentiality and countable tightness on the space \(({\mathbf C}(X),\tau_{\omega})\) of continuous real-valued functions in the properties of a topological space \(X\) with the fine topology \(\tau_{\omega}\). The main result is that the following are equivalent for a Tychonoff space \(X\), (1) \(X\) is pseudocompact; (2) \(({\mathbf C}(X), \tau_{\omega})\) is completely metrizable; (3) \(({\mathbf C}(X), \tau_\omega)\) has a countable tightness, which generalizes a theorem by G. Di Maio, L. Holá, D. Holý and R. A. McCoy [Topology Appl., 86, No.2, 105-122 (1998; Zbl 0940.54023)]. Some related results on \({\mathbf C}(X)\) with the graph topology and the Krikorian topology are obtained.
Reviewer: Shou Lin (Fujian)

54C35 Function spaces in general topology
54C05 Continuous maps
54E35 Metric spaces, metrizability
fine topology