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.
The key notions on which the dimension theory is based are the “minimal extensions of continua”, or “joins”, and the “splittings of continua over subcontinua”.
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REFERENCES
Buekenhout, F. (Ed.): 1995, Handbook of Incidence Geometry, North Holland Pub. Co.
Gerla, G.: 1985, ,Geometry without Points', Amer.Math. Monthly, 92, 707–71.
Gerla, G.: 1990, ,Pointless Metric Spaces', J. Symb. Logic 55, 207–711.
Gerla, G.: 1995, ,Pointless Geometries, in: F. Buekenhout (ed.), Handbook of Incidence Geometry, North Holland Pub. Co., pp. 1,015–1,031.
Grzegorczyk, A.: 1960, ,Axiomatization of Geometry without Points', Synthese 7, 228–235.
Huntington, V. E.: 1913, ,A Set of Postulates for Abstract Geometry', Math. Annalen 73, 522–559.
Hurewicz, W. and Wallman, H.: 1948, Dimension Theory, Princeton Univ. Press, p. 3.
Menger, K.: 1940, ,Topology without Points', The Rice Institute Pamphlet 27, 80–107.
Poincaré, A.: 1912, Revue de Metaphysique et de Morale, 1912, p. 486.
Munkress, J. R.: 1975, General Topology, Prentice Hall, p. 156.
Tarski, A.: 1926, ,Foundations of the Geometry of Solids, in: Logic, Metamathematics, Semantics (Papers from 1923–1938), Oxford University Press, 1956.
Tzouvaras, A.: 1993, ,Significant Parts and Identity of Artifacts', Notre Dame J. Formal Logic 34, 445–452.
Tzouvaras, A.: 1995, ,Worlds of Homogeneous Artifacts', Notre Dame J. Formal Logic 36, 454–474.
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Tzouvaras, A. The Order Structure of Continua. Synthese 113, 381–421 (1997). https://doi.org/10.1023/A:1005094430329
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DOI: https://doi.org/10.1023/A:1005094430329