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An isomorphical classification of function spaces of zero-dimensional locally compact separable metric spaces. (English) Zbl 0684.54011
Let \(C_ p(x)\) and \(C_ k(X)\) be the spaces of continuous real-valued functions on X with the pointwise convergence and compact-open topologies, respectively. C. Bessaga and A. Pełczyński [Stud. Math. 19, 53-62 (1960; Zbl 0094.303)] classified when \(C_ k(X)\) and \(C_ k(Y)\) are linearly homeomorphic for X and Y 0-dimensional compact metric spaces.
Now the authors do the same thing for \(C_ p(X)\) and \(C_ p(Y)\). Instead of using Banach spaces (which they could have), they use transfinite induction arguments. The authors also extend their results to locally compact spaces. In particular, they prove that if X and Y are 0- dimensional locally compact separable metric spaces then the following are equivalent:: (1) \(C_ p(X)\) is linearly homeomorphic to \(C_ p(Y)\). (2) \(C_ k(X)\) is linearly homeomorphic to \(C_ k(Y)\). (3) Exactly one of the following is true: (a) X and Y are both finite and have the same cardinality. (b) X is homeomorphic to [1,\(\alpha\) ] and Y is homeomorphic to [1,\(\beta\) ], where \(\alpha\) and \(\beta\) are countable infinite ordinals such that \(\max \{\alpha,\beta \}<(\min \{\alpha,\beta \})^{\omega}\). (c) X and Y are both uncountable and compact. (d) \(X=\oplus^{\infty}_{n=1}X_ n\) and \(Y=\oplus^{\infty}_{n=1}Y_ n\), where for each n, \(X_ n\) and \(Y_ n\) either both satisfy (a) or both satisfy (b) or both satisfy (c).
As shown by A. V. Arkhangel’skij [Soviet Math. Dokl. 25, No.3, 852- 855 (1982); translation from Dokl. Akad. Nauk SSSR 264, 1289-1292 (1982; Zbl 0522.54015)], if X and Y are any two metric spaces such that \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomoprhic, then so are \(C_ k(X)\) and \(C_ k(Y)\). So the authors establish the converse of this for 0- dimensional locally compact separable metric spaces. The 0-dimensionality is necessary for this converse, even if the spaces are compact.

54C35 Function spaces in general topology
46E10 Topological linear spaces of continuous, differentiable or analytic functions
57N17 Topology of topological vector spaces
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