Approach spaces: the missing link in the topology-uniformity-metric triad.

*(English)*Zbl 0891.54001
Oxford Mathematical Monographs. Oxford: Clarendon Press. x, 253 p. (1997).

The category AP of approach spaces (defined by the author about 10 years ago) contains nicely embedded categories of topological spaces, uniform spaces, and metric spaces with contractions. Mainly the last mentioned embedding makes the category AP different from other categories comprising Top, Unif, etc. An approach space is a pair \((X,\delta)\), where \(\delta\) is a real-valued function on \(X\times \exp X\) measuring a “distance” from points to subsets and satisfying certain natural axioms (several other equivalent descriptions of approach spaces are given). Morphisms \(f:(X,\delta)\to (X',\delta')\) are maps \(f:X\to X'\) having the property \(\delta'(f(x),f(A))\leq\delta(X,A)\) for all \(x\in X, A\subset X\), i.e., contractions. The category AP is a topological one. The three chapters 2, 3 and 4 contain properties of the embeddings of Top, Unif and Metr – e.g., Top is both bireflective and bicoreflective in AP. In the next Chapter, certain natural approach structures are described on function spaces, spaces of measures, hyperspaces, and others. Chapter 6 shows how approach structures allow to refine some known properties to get, e.g., measure of compactness, measure of connectedness (giving the usual connectedness in Top and the Cantor connectedness in Metr). Completions and compactifications of approach spaces are studied in the last two chapters; the \(\beta\)-compactification corresponds to the Samuel-Smirnov compactification. For convenience of readers, two short surveys are added of theory of categories and of some concrete structures (like metric spaces and their generalizations, convergence spaces, uniform spaces, proximity spaces).

Reviewer: M.Hušek (Praha)