##
**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.

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.