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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)

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