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On the Extension of the Pflastersatz

Published online by Cambridge University Press:  24 October 2008

B. Kaufmann
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Extract

In this paper I prove some new Pflastersätze for r−dimensional sets in the n−dimensional Euclidean space Rn belonging to one and the same family, in which Lebesgue's fundamental lemma represents a “0-dimensional” case out of r + 1 possible cases of dimensions 0, 1, 2, …, r.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 2 , May 1936 , pp. 238 - 247
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

† See Lebesgue, , Math. Annalen 70 (1911), 166;CrossRefGoogle ScholarUrysohn, P., Fund. Math. 8 (1926), 292;CrossRefGoogle ScholarAlexandroff, P., Math. Annalen, 98 (1926), 489;Google Scholar and Menger, K., Dimensionetheorie, Chap. v.Google Scholar

‡ A general r−dimensional manifold is an r−dimensional set (irreducibly) linked with an (n−r+1)-cycle rel a neighbourhood U.

† See Kaufmann, “On infinitesimal properties of closed sets of arbitrary dimension”, to appear shortly in Annals of Math. This paper is cited throughout as I.P. See also Comptes Rendus, 201 (1935), 416–418.Google Scholar This theory involves some fundamental results due to Alexandroff, , “Dimensionstheorie”, Math. Annalen, 106 (1932), 161–238.CrossRefGoogle Scholar

‡ Dimensions are understood in the sense of Brouwer (see Alexandroff, “Dimensionstheorie”, loc. cit.). The case r = 0 is trivial and is excluded from the proof.

§ In the sense of Alexandroff; we also I.P. III.

† If a cycle z (mod 0) bounds a complex in U, we write z ∼ 0 in U (for z ≃ 0 in U), and z ≁ 0 otherwise. All homologies linking relations, dissections, etc. are understood rel a spherical neighbourhood U (in the sense of Lefschetz).

‡ To simplify our expressions we include BT=F in the series of dissections. By an infinite cycle we mean a “Vollzyklus” in Alexandroff's sense (not to be confused with infinite cycles in the sense of Lefschetz).

† It may be pointed out here (in connection with the paper quoted above) that it is sufficient for our purpose to apply ε modifications with respect to the set B 0 only.

† See I.P. IV. 4, 5.

‡ We write here {sn−r+j−1} for {zn−r+j−1}.

† The extension of the results of this paragraph to arbitrary closed sets wi11 be added in the continuation of this paper.

† See I.P. I. In the case of h = r we may expect the corresponding Pflastersätze to be valid for Risspunkte of a higher order r+l in the sense that the arbitrarily all manifolds of a sequence defining a manifold point contain inner points of r+l parts of the given subdivision.

† See I.P. I and IV.

‡ It may be mentioned with regard to the set of points theory of dimensions that there was no evidence of the existence even of a single point of that kind in a closed set.