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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.

MSC:
 54C35 Function spaces in general topology 54-02 Research exposition (monographs, survey articles) pertaining to general topology 57N17 Topology of topological vector spaces 57N20 Topology of infinite-dimensional manifolds