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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
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