Effective topological spaces. I: A definability theory
type: Item
instance of: scholarly article
title: Effective topological spaces. I: A definability theory
zbMATH Open document ID: 0569.03018
DOI: 10.1016/0168-0072(85)90033-8
author: Iraj Kalantari
author: Galen Weitkamp
published in: Annals of Pure and Applied Logic
publication date: 1985-01-01T00:00:00Z
review text: This is the first episode of a three parts study intending to introduce an arithmetization and a first-order logic for the theory of topological spaces. The main subject of the work under review is the definition of satisfaction for formulas expressing topological properties. The main idea is that ''in computing practice one rarely has perfect information about a point but at best has an approximation in the form of a neighborhood''. Thus for each topological space X a first-order language without equality \(L_ X\) is associated such that the open subsets of the n-fold cartesian product \(X^ n\) under the product topology are designated by the n-ary predicate symbols of \(L_ X\). For such a predicate P, ''P(\(\alpha)\) is true'' means that \(\alpha\) is contained in the regular open set determined by P. This definition is adequately extended for the logical connectives and quantifiers of \(L_ X\). Some important logical rules of deduction, such as modus ponens and syllogism, are proved. It is clarified what it means for a formula to define a set and then there are found criteria for deciding whether a formula is a definition or not. In conclusion the present approach seems to be more adequate and general than the classical effective presentation through the rationals of the real line topology.
Mathematics Subject Classification ID: 03D45
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 3908421
zbMATH Keywords: first-order theory
zbMATH Keywords: arithmetization
zbMATH Keywords: theory of topological spaces
zbMATH Keywords: formulas expressing topological properties
reviewed by: Marius Zimand
MaRDI profile type: Publication
full work available at URL: 0168-0072(85)90033-8
OpenAlex ID: W2067792740
cites work: Sostituzioni inverse
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description: scientific article; zbMATH DE number 3908421