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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.

MSC:
 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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