Spaces in which there exist contiguous points




Klipple, Edmund Chester

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R. L. Moore has formulated a set of axioms in terms of the undefined notions point, region, and "contiguous to". These axioms (Axioms A, B, 0, 1, and 2 of this paper) serve as a sufficient basis for the proofs of a considerable number of theorems of ordinary point set theory. For example, in any space satisfying these axioms, the Borel-Lebesgue Theorem holds true. Also it is true that if A and B are two distinct points of a connected domain D, then there exists a simple continuous arc from A to B lying wholly in D. Nevertheless there exist spaces satisfying these axioms in which an arc may contain only a finite number of points and in which a region may consist of a finite number of points. In the present paper Moore's set of axioms and an additional axiom (Axiom 3 below) are used in proving certain theorems about simple closed curves. The principal theorem of the paper is an analogue of the theorem of the ordinary Euclidean plane which states that if each of the simple closed curves J and C encloses the point P, then there exists a simple closed curve Q which is a subset of J + C and whose interior contains P and is enclosed by both J and C. Roughly speaking, Axiom 3 restricts the space to two dimensions. Furthermore, the ordinary Euclidean plane is a space which satisfies all the axioms

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