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The Cantor manifolds lying on a closed surface

Published online by Cambridge University Press:  24 October 2008

Boris Kaufmann
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Extract

In recent years Cantor manifolds have assumed great importance in topology. This is due chiefly to the work of Urysohn and to the later work of Alexandroff. Urysohn, in the paper in which he introduced the theory of dimensions, showed that a 2-dimensional closed surface is a 2-dimensional Cantor manifold. Alexandroff has not only extended this result to n dimensions but has also established the existence on the cutting set Z of the surface F of generalized cycles homologous to zero on the various parts of F − Z. These results show that the deepest structural characteristics of the closed surface come to light when one dissects it.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 4 , October 1934 , pp. 428 - 444
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

† Urysohn, P., Fundamenta Mathematicae, 7 (1925), 29–137.CrossRefGoogle Scholar

An n-dimensional closed surface is a bounded closed set F in an (n + 1)-dimensional Euclidean space R such that R − F consists of two domains G 1, G 2 each having F for its boundary. A compact n-dimensional point-set F is called an n-dimensional Cantor manifold if every closed subset Z of F, such that F − Z is not connected, is of dimension n − 1 at least.

‡ Alexandroff, P., Annals of Mathematics (2), 30 (1929), 101–187.CrossRefGoogle Scholar

§ Hurewicz, W. and Menger, K., Math. Ann. 100 (1928), 618–633.CrossRefGoogle Scholar

† A cut F of a connected region G we call regular (after Urysohn) if F is the common boundary of two connected parts of G − F. Alexandroff, loc. cit. p. 10, uses the phrase absolute Gebietsgrenze if F is the common boundary of all components.

‡ See Veblen, O., Analysis situs (1931).Google Scholar

† If a convex polygon lying on has on its boundary a segment l of the boundary of there may be an interior point of l which is a vertex of a convex polygon drawn on another simplex of K2. Any such points are to be regarded as vertices of the first polygon as well in the final triangulation.

† See note † on p. 429.

‡ See note † on p. 428.

§ Our result includes that of Urysohn, and we have simplified his proof.

∥ Loc. cit. p. 101.

† Otherwise we could, for certain arbitrarily large values of ν, draw a curve C ν joining q 1 to q 2 on Πν without entering P n. We may obviously suppose that these curves have a common point on q, and then it follows that the upper limiting set C is a continuum. But C lies on F − Z and joins q 1 to q 2, and this leads to a contradiction.

† For the case of the plane a similar result was given by Menger, following a suggestion of O. Schreier: in principle the above result contains nothing new. Its application—after a further extension—is indispensable in § 4.

† It is not difficult to show that it cannot be empty.

† In the sense of § 3.

† Otherwise we could find a sequence of continua lying on C, having no points in common, and all of diameter ≥ ε > 0. The limit set would be 1-dimensional, and would lie on Z *: which is impossible since Z * is 0-dimensional.