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Completing Quasi-metric Spaces – an Alternative Approach. (English) Zbl 1032.54013

It is known that approach spaces, a common supercategory of TOP and MET, are extremely useful when solving problems having both topological and uniform aspects. As the authors explain: “Quasi-metric spaces have, precisely because they are the key-tool expressing asymmetry, arisen in many parts of mathematics in which precise order plays an important role: e.g. in theoretical computer science, ... it is exactly the completeness of certain quasi-metric structures which theoretically guarantees that certain constructions stop.”
For information about quasi-metric spaces and their applications the reader is referred to the recent survey by H.-P.A.Künzi [Handbook of the History of General Topology, Vol. 3, (Kluwer Academic Publishers, Dordrecht) (2001; Zbl 1002.54002)].
An approach space is said to be \(T_0\) when its topological coreflection is \(T_0\). In this paper, for each \(T_0\) approach space \(X\) its completion \(\hat X\) is constructed in such a way that when \(X\) satisfies additional natural conditions (limit regularity), then \(\hat X\) has good categorical properties and, since quasi-metric spaces can be viewed in a natural way as approach spaces, the construction yields a completion for quasi-metric spaces having good categorical properties, too.
The completion theory developed agrees with the usual completion theory for uniform approach spaces, and hence it extends the usual theory for metric spaces.
Sample results: If \(X\) is \(T_0\), then \(X\) is densely embedded in \(\hat X\) and \(\hat X\) is \(T_0\), too. The full subcategory of limit-regular approach spaces is bireflective in the category AP of approach spaces. The full subcategory of \(T_0\) limit-regular approach spaces is epireflexive in AP. In the category of \(T_0\) limit-regular approach spaces, the subcategory of complete spaces is epireflective, and the reflection is the completion.

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54E99 Topological spaces with richer structures
18B30 Categories of topological spaces and continuous mappings (MSC2010)

Citations:

Zbl 1002.54002
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