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Frames. (English) Zbl 1063.06006

Hazewinkel, M. (ed.), Handbook of algebra. Volume 3. Amsterdam: Elsevier (ISBN 0-444-51264-0/hbk). 791-857 (2003).
In the space of fewer than 70 pages allowed by the editors of this Handbook, it is clearly impossible to do justice to all aspects of the theory of frames or locales (also known as ‘pointfree topology’). But the author has done a remarkably good job of outlining the basic theory for an audience having no previous acquaintance (or at least not presumed to have any) with frames. He concentrates on the ‘classical’ theory, and does not discuss the connections of the subject with constructive logic and topos theory, or its applications in fibrewise topology; but what he does cover is surprisingly comprehensive, and succeeds very well in giving the reader a feel both for the ways in which classical locale theory is substantially similar to classical point-set topology, and also for the ways in which it not only generalizes but improves upon point-set topology. Considerations of space mean that some proofs have had to be omitted, and others are given in rather telegraphic style, but there are numerous pointers to the extensive bibliography (over 150 items) for readers who wish to see more detail.
Reviewer’s remark: Since the author refers to his own lack of expertise in the history of the subject, it is perhaps worth mentioning one reference which is not in the bibliography: the reviewer contributed a chapter on the history of locale theory to the third volume of the Handbook of the history of general topology, Hist. Topol. 3, 835–851 (2001; Zbl 1001.54001).
For the entire collection see [Zbl 1052.00009].

MSC:

06D22 Frames, locales
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54A05 Topological spaces and generalizations (closure spaces, etc.)
54B30 Categorical methods in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
06-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to ordered structures
54-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to general topology

Citations:

Zbl 1001.54001
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References:

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