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Generalized metric spaces. (English) Zbl 0555.54015

Handbook of set-theoretic topology, 423-501 (1984).
[For the entire collection see Zbl 0546.00022.]
From the introduction: ”What is a ’generalized metric space’? There is no precise definition. Any class of spaces defined by a property possessed by all metric spaces could technically be called a class of generalized metric spaces. But the term is meant for classes which are ’close’ to metrizable spaces in some sense: they usually possess some of the useful properties of metric spaces, and some of the theory or techniques of metric spaces carries over to these wider classes. Sometimes they can be used to characterize the images or pre-images of metric spaces under certain kinds of mappings. They often appear in theorems which characterize metrizability in terms of weaker topological properties. To be most useful, they should be ’stable’ under certain topological operations, e.g., finite or countable products, closed subspaces, and perfect (closed, with compact point-inverses) mappings.”
This is an excellent survey for both the beginner and the worker in the field. The author starts with basic metrization theory and developable spaces. Proofs are given of most of the important culminating theorems involving generalized metric spaces and metrization and developability of the past 30 years for the topics above and even some older theorems as the Moore metrization theorem. To the reviewer’s knowledge, the first proof in English is presented of Ponomarev’s nice result that a space has a point countable base iff it is the continuous s-image of a metric space. Many other important mapping theorems are also proved. The paper has many important examples and unsolved problems are presented throughout the paper. The survey is well-organized and has an index of definitions indicating in what theorems the defined terms appear in. The paper is an invaluable guide to the literature with an extensive bibliography.
Reviewer: C.E.Aull

MSC:

54E35 Metric spaces, metrizability
54-02 Research exposition (monographs, survey articles) pertaining to general topology
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54E20 Stratifiable spaces, cosmic spaces, etc.
54E25 Semimetric spaces
54E30 Moore spaces

Citations:

Zbl 0546.00022