Region-based topology
This is a very original investigation in the philosophy of topology. Its basic ideas are not totally unheard of; the author cites anticipations by K. Menger and A. Tarski, and a topologist notes more technical bases in \textit{M. H. Stone's} apparatus of Boolean maps [Trans. Am. Math. Soc. 41, 375-481 (1937; Zbl 0017.13502)]\ and in \textit{A. M. Gleason's} projective covers of locally compact Hausdorff spaces [Ill. J. Math. 2, 482-489 (1958; Zbl 0083.17401)]. The author must first introduce the primitive terms for his region-based topology, with intuitive explanations and also, of course, precise axioms. The regions which underlie everything are members of some Boolean algebra. Almost exactly in the middle of the paper (p. 279) the author says ` ` In the light of' ' \ the Main Theorem ` ` it seems appropriate to ignore from now on region-based topologies that are based on incomplete Boolean algebras' ' . The Main Theorem says that complete region-based topologies correspond reversibly (up to isomorphism/homeomorphism) with locally compact Hausdorff spaces. So the point of the first half of the paper is to describe those spaces in the ` ` regional' ' \ language. There is no hope of presenting that language in this review. The first three predicates for regions are binary \textit{connection}, which is like proximity, and unary \textit{coherence}, which is exactly (in the complete case, at least) connectedness, and \textit{convexity}, which is a way of having a nice boundary. The second half of the paper consists of some 16 pages on familiar properties such as local connectedness and the Lindelöf property, in regional language, and on something far more novel and tentative which I shall return to, and also 11 pages on \textit{functions} (to a topologist, morphisms). The author' s category is not the category of locally compact Hausdorff spaces and continuous mappings. The results do not seem conclusive, but the reviewer suspects that the author' s intuition has misled him here; the morphisms (\textit{mereological mappings}) certainly need to prove themselves. The results given on their properties are substantial. The tentative development mentioned above results from omitting the last of the ten axioms needed for the main theorem, which ` ` postulates a certain kind of divisibility of the space' ' . It appears to be like the Russell-Whitehead Axiom of Reducibility at least in being very much wanted for technical reasons, the reasons here turning on the management of ultrafilters: getting points from regions. Examples illustrate the difficulties that its absence causes, and a one-page appendix introduces further apparatus which may lead to overcoming them.
scientific article; zbMATH DE number 1044154