On topological and linear equivalence of certain function spaces.

*(English)*Zbl 0755.54007
CWI Tracts. 86. Amsterdam: Centrum voor Wiskunde en Informatica. 201 p. (1992).

This book gives a survey, including recent work by the authors and others, of the theory of topological and linear equivalence of function spaces. The primary function spaces considered are the spaces \(C_ p(X)\) and \(C_ 0(X)\) of continuous real-valued functions on a Tychonoff space \(Y\) under the topologies of pointwise convergence and compact-open, respectively. Also considered are \(C_ p^*(X)\) and \(C_ 0^*(X)\), where the asterisk indicates bounded functions.

Nagata’s theorem, that \(C_ p(X)\) and \(C_ p(Y)\) are topologically isomorphic as topological rings if and only if \(X\) and \(Y\) are homeomorphic, suggests the two general problems that generate this theory. These problems are: (1) if \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic (or just homeomorphic), which topological properties on \(X\) are necessarily preserved on \(Y\); and (2) under what conditions on \(X\) and \(Y\) are \(C_ p(X)\) and \(C_ p(Y)\) linearly homeomorphic (or just homeomorphic)? These problems also have their analogs for \(C_ 0(X)\), \(C_ p^*(X)\) and \(C_ 0^*(X)\).

The first chapter gives a nice exposition of the basic tools used in attacking these problems. Some of these tools, in the case of \(C_ p(X)\) and \(C_ 0(X)\), are due to Arkhangelskij; and others are used without proof in papers by Pavlovskij, Pestov and others. An example of this latter kind is: if \(L(X)\) is the topological dual of \(C_ p(X)\), then \(X\) can be naturally embedded as a closed subspace of \(L(X)\) which is in fact a Hamel basis for \(L(X)\); and that \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic if and only if \(L(X)\) and \(L(Y)\) are linearly homeomorphic. The first chapter closes with a selection of theorems that can be established from these tools. For example, if \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic, then whenever one of \(X\) or \(Y\) is compact the other must also be compact (i.e., compactness is said to be preserved by \(\ell_ p\)- equivalence).

In the second chapter the authors give a complete isomorphical classification of the function spaces \(C_ p(X)\) and \(C_ 0(X)\) (as topological vector spaces) where \(X\) is a member of one of the classes: compact zero-dimensional metric spaces, compact ordinal spaces, \(\sigma\)- compact ordinal spaces, and separable locally compact zero-dimensional metric spaces. This extends and completes the classifications given by Bessaga and Pełczyński for \(C_ 0(X)\) where \(X\) is a compact zero-dimensional metric space and by Kislyakov for \(C_ 0(X)\) where \(X\) is a compact ordinal space. A special feature in this chapter is that, because of the extensive use of ordinal numbers, there is a section covering properties of ordinal numbers, including ordinal arithmetic and prime components. There is also a section on derived sets and scattered spaces. A third section covers factorizing function spaces, which is a useful tool to help understand the linear topological structure of function spaces.

The third chapter contains the material on topological equivalence of function spaces. It starts with a section covering some topics from infinite-dimensional topology. These are used to give two different proofs of the topological classification of \(C_ p(X)\) where \(X\) is a nonlocally compact countable metric space (every two such function spaces are homeomorphic); the second proof of which follows the method used by van Mill to prove the corresponding result for \(C_ p^*(X)\), and which uses theorems by Toruńczyk.

The final chapter deals with isomorphical classification of \(C_ p(X)\) where \(X\) is any separable zero-dimensional metric space that is not locally compact. The main tool here is that of \(\ell_ p\)- equivalent pairs. This is used to establish the isomorphical classification of \(C_ p(X)\) where X is any countable metric space with scattered height less than or equal to \(\omega\). But in the next two sections, the authors give examples and arguments that indicate that an isomorphical classification for function spaces \(C_ p(X)\) for all countable metric spaces \(X\) may be beyond reach. The last two sections give some partial results for the spaces \(C_ 0(X)\) and \(C^*_ p(X)\).

This book is reasonably self-contained, although it does use a few major theorems from other areas such as infinite-dimensional topology. The authors pose a number of questions for further research, and even include one section that discusses a conjecture. The book brings the reader to the forefront of the theory of topological and linear equivalence of function spaces.

Nagata’s theorem, that \(C_ p(X)\) and \(C_ p(Y)\) are topologically isomorphic as topological rings if and only if \(X\) and \(Y\) are homeomorphic, suggests the two general problems that generate this theory. These problems are: (1) if \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic (or just homeomorphic), which topological properties on \(X\) are necessarily preserved on \(Y\); and (2) under what conditions on \(X\) and \(Y\) are \(C_ p(X)\) and \(C_ p(Y)\) linearly homeomorphic (or just homeomorphic)? These problems also have their analogs for \(C_ 0(X)\), \(C_ p^*(X)\) and \(C_ 0^*(X)\).

The first chapter gives a nice exposition of the basic tools used in attacking these problems. Some of these tools, in the case of \(C_ p(X)\) and \(C_ 0(X)\), are due to Arkhangelskij; and others are used without proof in papers by Pavlovskij, Pestov and others. An example of this latter kind is: if \(L(X)\) is the topological dual of \(C_ p(X)\), then \(X\) can be naturally embedded as a closed subspace of \(L(X)\) which is in fact a Hamel basis for \(L(X)\); and that \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic if and only if \(L(X)\) and \(L(Y)\) are linearly homeomorphic. The first chapter closes with a selection of theorems that can be established from these tools. For example, if \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic, then whenever one of \(X\) or \(Y\) is compact the other must also be compact (i.e., compactness is said to be preserved by \(\ell_ p\)- equivalence).

In the second chapter the authors give a complete isomorphical classification of the function spaces \(C_ p(X)\) and \(C_ 0(X)\) (as topological vector spaces) where \(X\) is a member of one of the classes: compact zero-dimensional metric spaces, compact ordinal spaces, \(\sigma\)- compact ordinal spaces, and separable locally compact zero-dimensional metric spaces. This extends and completes the classifications given by Bessaga and Pełczyński for \(C_ 0(X)\) where \(X\) is a compact zero-dimensional metric space and by Kislyakov for \(C_ 0(X)\) where \(X\) is a compact ordinal space. A special feature in this chapter is that, because of the extensive use of ordinal numbers, there is a section covering properties of ordinal numbers, including ordinal arithmetic and prime components. There is also a section on derived sets and scattered spaces. A third section covers factorizing function spaces, which is a useful tool to help understand the linear topological structure of function spaces.

The third chapter contains the material on topological equivalence of function spaces. It starts with a section covering some topics from infinite-dimensional topology. These are used to give two different proofs of the topological classification of \(C_ p(X)\) where \(X\) is a nonlocally compact countable metric space (every two such function spaces are homeomorphic); the second proof of which follows the method used by van Mill to prove the corresponding result for \(C_ p^*(X)\), and which uses theorems by Toruńczyk.

The final chapter deals with isomorphical classification of \(C_ p(X)\) where \(X\) is any separable zero-dimensional metric space that is not locally compact. The main tool here is that of \(\ell_ p\)- equivalent pairs. This is used to establish the isomorphical classification of \(C_ p(X)\) where X is any countable metric space with scattered height less than or equal to \(\omega\). But in the next two sections, the authors give examples and arguments that indicate that an isomorphical classification for function spaces \(C_ p(X)\) for all countable metric spaces \(X\) may be beyond reach. The last two sections give some partial results for the spaces \(C_ 0(X)\) and \(C^*_ p(X)\).

This book is reasonably self-contained, although it does use a few major theorems from other areas such as infinite-dimensional topology. The authors pose a number of questions for further research, and even include one section that discusses a conjecture. The book brings the reader to the forefront of the theory of topological and linear equivalence of function spaces.

Reviewer: R.A.McCoy (Blacksburg)