The order structure of continua

Synthese 113 (3):381-421 (1997)
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Abstract

A continuum is here a primitive notion intended to correspond precisely to a path-connected subset of the usual euclidean space. In contrast, however, to the traditional treatment, we treat here continua not as pointsets, but as irreducible entities equipped only with a partial ordering ≤ interpreted as parthood. Our aim is to examine what basic topological and geometric properties of continua can be expressed in the language of ≤, and what principles we need in order to prove elementary facts about them. Surprisingly enough ≤ suffices to formulate the very heart of continuity (=jumpless and gapless transitions) in a general setting. Further, using a few principles about ≤ (together with the axioms of ZFC), we can define points, joins, meets and infinite closeness. Most important, we can develop a dimension theory based on notions like path, circle, line (=one-dimensional continuum), simple line and surface (=two-dimensional continuum), recovering thereby in a rigorous way Poincaré's well-known intuitive idea that dimension expresses the ways in which a continuum can be torn apart. We outline a classification of lines according to the number of circles and branching points they contain. The ordering (C,≤) is a topped and bottomed, atomic, almost dense and complete partial ordering, weaker than a lattice. Continuous transformations from C to C are also defined in a natural way and results about them are proved.

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Athanassios Tzouvaras
Aristotle University of Thessaloniki (PhD)

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Pointless metric spaces.Giangiacomo Gerla - 1990 - Journal of Symbolic Logic 55 (1):207-219.
Significant parts and identity of artifacts.Athanassios Tzouvaras - 1993 - Notre Dame Journal of Formal Logic 34 (3):445-452.
Worlds of Homogeneous Artifacts.Athanassios Tzouvaras - 1995 - Notre Dame Journal of Formal Logic 36 (3):454-474.

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