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On the extension of the Pflastersatz (Part II)

Published online by Cambridge University Press:  24 October 2008

B. Kaufmann
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Extract

In the first part of this paper I announced some new Pflaster theorems for arbitrary r-dimensional closed sets lying in the Euclidean space Rn. In § III I proved them for the special case of an r-dimensional closed set F linked (rel a neighbourhood U) with an (n – r – 1)-dimensional spherical cycle. I shall now prove these theorems in the general case of a quite arbitrary closed set F.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 1 , March 1937 , pp. 13 - 20
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

* “On the extension of the Pflastersatz”, Proc. Camb. Phil. Soc. 32 (1936), 238.Google Scholar I shall quote this paper as “Part I”.

† The proofs of these theorems are given in my paper “On infinitesimal properties of closed sets of arbitrary dimension” (to appear in Annals of Math.). I shall quote this paper as “IP”.

‡ See also § 1, Part I. We say also that the system forms an r-fold dissection of the set A r in dual correspondence with the system of r-fold expansions of the cycle z n−r−1.

* An ε subdivision F = ∑F i of a set F is called canonical, if any k parts (k = 2, 3, …, r + 2) have an at most (r − k + 1)-dimensional set in common. The existence of canonical subdivisions of a closed set for any ε is well known.

† This construction, which is completely described in § II (IP), is based on Lemma 1 (§ 1) on the expansion of cycles.

* See the proof of the inductive Phragmen-Brouwer theorem in § 2 (IP). The Phragmen-Brouwer-Alexandroff theorem is a special case of our Theorem PB (Part I, § I).

* The proof of (v)j+1 obviously applies verbatim in the case j = 0.