Concerning R.L. Moore's axiom 5₁




Jones, F. Burton

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In any space satisfying Axioms 0, 1, 2, 3, 4, and 5 of R.L. Moore's Foundations of Point Set Theory a large body of Plane Analysis Situs theorems hold true. Nevertheless, not every such space is a subset of the plane even if completely separable. However, Moore has shown that if such a space satisfies certain additional axioms, it is a subset of a plane. This problem has also been studied by Leo Zippin and, more recently, by J.H. Roberts. In this paper the problem is again attacked, and the treatment emphasizes the rather peculiar role played by connectedness. As a matter of fact, connectedness enters into all of the axioms used by the author except Axioms 0, 1, and 6. In Part I a certain space is studied which, although not necessarily a subset of a plane, nevertheless, possesses many of the properties of a plane. However, if the space of Part I is completely separable or metric, it is shown to be a subset of a plane or a sphere. Definitions of terms peculiar to the treatment are given in the text. For the definition of terms not defined, the reader is refered to R.L. Moore's Foundations of Point Set Theory. Many of the notational conventions of this work are also used here without explanation.