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$$\lambda$$-topologies on function spaces. (Russian. English summary) Zbl 1068.22003
Fundam. Prikl. Mat. 9, No. 2, 3-56 (2003); translation in J. Math. Sci., New York 131, No. 4, 5701-5737 (2005).
Let $$X$$ be a Tikhonov space and $$C(X)$$ be a linear space of all continuous real-valued functions over $$X$$. The set $$A$$ is said to be bounded in $$X$$ if any function $$f\in C(X)$$ is bounded on $$A$$. Let $$\lambda$$ be a family of bounded subsets in $$X$$. The topology of uniform convergence on the elements $$\lambda$$ is called a $$\lambda$$-topology. The paper contains a survey of the author’s results devoted to the space $$C_{\lambda}(X)$$ of all continuous real-valued functions on $$X$$ endowed with arbitrary $$\lambda$$-topologies. The preferences are given to the following subjects: cardinal functions, locally convex properties, weak and strong topologies, dual spaces, lattices of $$\lambda$$-topologies, completeness.

##### MSC:
 22A10 Analysis on general topological groups 54C30 Real-valued functions in general topology 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54C35 Function spaces in general topology 54E50 Complete metric spaces
##### Keywords:
functional spaces; $$\lambda$$-topologies