Concerning upper semi-continuous collections-Reference-Cited by-同舟云学术

Concerning upper semi-continuous collections

Published:1929-12 Issue:1 Volume:36 Page:81-88
ISSN:0026-9255
Container-title:Monatshefte für Mathematik und Physik
language:de
Short-container-title:Monatsh. f. Mathematik und Physik

Author:

Moore R. L.

Publisher

Springer Science and Business Media LLC

Subject

General Mathematics

Link

http://link.springer.com/content/pdf/10.1007/BF02307605.pdf

Reference18 articles.

1. Concerning upper semi-continuous collections of continua, Transactions of the American Mathematical Society, vol. 27 (1925), pp. 416–428.

2. A cyclic subset of a continuous curveM is a subsetK ofM such that every two points ofK belong to some simple closed curve which is a subset ofK and a cyclic subset ofM is said to be maximal if it is not a proper subset of any other cyclic subset ofM. See the following papers by G. T. Whyburn: Cyclically connected continuous curves. Proceedings of the National Academy of Sciences, vol. 13 (1927), pp. 31–38; Some properties of continuous curves, Bulletin of the American Mathematical Society, vol. 33 (1927), pp. 305–308; Concerning the structure of a continuous curve, American Journal of Mathematics, vol. L (1928), pp. 167–194. The term maximal cyclic subset as used here is more inclusive than Whyburns maximal cyclic continuous curve only in that a point may be a maximal cyclic subset ofM but no point is a cyclic continuous curve in the sense of Whyburn. On the other hand Whyburn's cyclic element is more inclusive than maximal cyclic subset in that a cut point ofM that belongs to a maximal cyclic continuous curve ofM is a cyclic element ofM but not a maximal cyclic subset ofM. This slight departure from Whyburn's terminology is made, in the present paper, largely for contextual reasons. A maximal cyclic subset ofM will be said to be degenerate if it consists of a single point.

3. Vietoris has shown that if the spaceS G is one dimensional then it is an acyclic continuous curve. Every such curve is a cactoid whose maximal cyclic subsets are all points. Cf. L. Vietoris, Über stetige Abbildungen einer Kugelfläche, Proceedings of the Royal Academy of Sciences of Amsterdam, vol. 29 (1926), pp. 443–453.

4. R. L. Moore, loc. cit. (1923), pp. 101–106, Theorem 26. While this theorem was established for a Euclidean spaceS ofn-dimensions, it is clear that (if the last definition of Page 417 is omitted) the same argument suffices to establish it for the case where the spaceS is a spherical surface or, indeed, any metric space which itself satisfies Axioms 1, 2, 4 and 51 and Theorem 4 of my Foundations article. That every space satisfying Axiom 1 and Theorem 4 is necessarily metric may be easily seen with the help of the Urysohn-Tychonoff theorem to the effect that every regular and perfectly separable Hausdorff space is metric. See P. Urysohn, Mathematische Annalen, vol. 94 (1925), pp. 309–315 and A. Tychonoff, Ibid. vol. 95 (1926), pp. 139–142. It has been shown by Chittenden that in order that a topological space in which there are no isolated points should be metric and separable it is necessary and sufficient that is should satisfy my Axiom 1. Cf. E. W. Chittenden, On the metrization problem and related problems in the theory of abstract sets, Bulletin of the American Mathematical Society, vol. 33 (1927), pp. 13–34.

5. Transactions of the American Mathematical Society, vol. 17 (1916), pp. 131–164.

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1. Proper minimal sets on compact connected 2-manifolds are nowhere dense;Ergodic Theory and Dynamical Systems;2008-06

2. History of Continuum Theory;Handbook of the History of General Topology;1998

3. The mathematical work of R. L. Moore: Its background, nature and influence;Archive for History of Exact Sciences;1982-03

4. Plane Topology;Dynamic Topology;1979

5. Robert Lee Moore, 1882-1974;Bulletin of the American Mathematical Society;1976