Region-based topology
type: Item
instance of: scholarly article
title: Region-based topology
zbMATH Open document ID: 0873.54001
DOI: 10.1023/A:1017904631349
author: Peter Roeper
published in: Journal of Philosophical Logic
publication date: 1997-10-16T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54H99
zbMATH DE Number: 1044154
zbMATH Keywords: region
zbMATH Keywords: ultrafilter
zbMATH Keywords: mereology
reviewed by: J. R. Isbell
MaRDI profile type: Publication
cites work: Q3206923
cites work: Q5721231
cites work: TOPOLOGY WITHOUT POINTS
cites work: Generalisation of first-order logic to nonatomic domains
cites work: Q5585437
cites work: Q4060097
full work available at URL: a:1017904631349
OpenAlex ID: W2057991286
description: scientific article; zbMATH DE number 1044154