|Title||Topological Function Spaces|
|Publisher||Kluwer Academic Publishers|
|Year of publication||1992|
|Reviewed by||S. Cobzas|
The book under review is concerned with the study of the space Cp(X) of all real-valued continuous functions on the topological space X in the topology of pointwise convergence. The main object of investigation is how the properties of the spaces X and Cp(X) are related. One of the greatest merits of the topology of pointwise convergence is the fact that it is the smallest of practically all natural topologies on C(X), and hence gives the largest amount of compacta.
The space Cp(X) can be considered as a topological space only, as a uniform space, as a topological group, as a linear (locally convex) topological space or as a topological ring. Corresponding to this structures on Cp(X) appears the corresponding problems for the topological space X. Namely, distinguish :
1. the supertopological properties of X , i.e. those properties characterized by the topological structure of Cp(X);
2. the u-properties of X, characterized by the uniform structure of Cp(X);
3. the linear topological properties of X, characterized by the linear topological structure of Cp(X);
4. The algebraic properties of X, characterised by the algebraic structure of Cp(X); and, finally
5. the ring properties of X, characterized by the topological ring properties of Cp(X).
A remarkable result in this direction is Nagata's theorem: If X and Y are Tikhonov spaces then the rings Cp(X) and Cp(Y) are topological isomorphic if and only if the spaces X and Y are homeomorphic (J. Nagata, Osaka Math. J. 1 (1949), 166-181).
A Tikhonov space is a completely regular space in which every finite set is closed. All the spaces considered in this book are taken to be Tikhonov spaces. The space Cp(X) is Cech complete if and only if X is discrete and countable, where Cech complete means that X is a set of a Gd-type in any Hausdorff compactification of it. For a metric space X, Cech completeness is equivalent to metrizability in a complete metric.
At the same time the ring Cp(X) is always algebraically isomorphic to the ring Cp(vX), where vX is the Hewitt-Nachbin compactification of X, showing that the algebraic properties of Cp(X) only are not sufficient to characterize the topological space X.
Another class of problems considered in the book are of the following type: For a given class P of topological spaces characterize the class H(P) of all spaces Y such that there exists X in P such that Y is homeomorphic to a subspace of Cp(X). In this direction, a special attention is paid to Eberlein compacta, i.e. compact subsets of Banach spaces in the weak topology, and to Lindelöf spaces and their extensions, to which is dedicated an entire chapter - Chapter IV, Lindelöf number properties for function spaces over compacta similar to Eberlein compacta, and properties of such compacta.
Beside these types of problems, relating X and Cp(X), it is expedient to study Cp(X) by itself, the particularities of the structure of the space Cp(X) implying specific relations between its topological properties, E. g. the space Cp(X) is paracompact if and only if it is Lindelöf; Cp(X) is always linearly and topologically isomorphic to <formula>, and other properies involving more sophisticated topological notions.
Written by an authoritative personality in the field, with deep and outstanding results in general topology, the book is very well written, contains a wealth of information, most of it scattered before through journal papers, some of them inaccessible to a large audience. The bibliography listed at the end of the book contains 159 items. The book contains also many open problems deserving further investigation on the part of the reader.
The book is printed in excellent typographical conditions, and we recommend it warmly to all interested in general topology, functional analysis and their applications.