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The Cantor Manifolds Lying on a Closed Surface. Part II

Published online by Cambridge University Press:  24 October 2008

H. D. Ursell
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Let F be a regular surface in ordinary space and let Z be a cut on F. Then there is on Z a point z which is the limit of a decreasing sequence of 2-dimensional Cantor manifolds lying on F.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 2 , April 1935 , pp. 183 - 194
Copyright
Copyright © Cambridge Philosophical Society 1935

References

† Kaufmann, , Proc. Camb. Phil. Soc. 30 (1934), 428.CrossRefGoogle Scholar

A regular cut of a connected region G is a closed subset F of G such that G − F is not connected and which is the common boundary (relative to G) of at least two connected parts of G − F. A regular surface is simply a regular cut of the whole space. A closed surface is a particular case of a regular surface; in a short note in the Quart. Journ. Math. we give reasons for dealing with the more general class of surface in this work. A cut on F is a closed subset Z such that F − Z is not connected. An h-dimensional Cantor manifold is an h-dimensional set F on which every cut is at least (h − 1)-dimensional.

† Kaufmann, Der lokale Dimensionsbegriff (to appear shortly).

‡ Menger, , Monatshefte f. Math. u. Phys. 33 (1923), 153.Google Scholar

§ Kaufmann, , Proc. Camb. Phil. Soc. 30 (1934), 428.CrossRefGoogle Scholar

† Alexandroff, , Annals of Mathematics, 30 (1928), 101.CrossRefGoogle Scholar We require the analogue for a region of the Verallgemeinerter Urysohnscher Satz on dissection of the whole space given by Alexandroff on p. 154 of this paper. This analogue for the case of a closed spherical region of space follows at once from the original theorem by considering a transformation which turns the whole boundary of the region into a single point. And this case is sufficient for all our purposes. The corresponding theorem for a region proper can also be deduced by this transformation; but we require the result that the dimension of a closed set (of more than one point) is never diminished by the removal of a single point from the set. And this result may easily be deduced from the work of Menger and Hurewicz, , Math. Ann. 100 (1928), 618.CrossRefGoogle Scholar These analogues are both contained (implicitly) in a later work of Alexandroff, , Math. Ann. 106 (1932), 161.CrossRefGoogle Scholar

† Alexandroff, loc. cit.

† Menger, and Hurewicz, , Math. Ann. 100 (1928), 618.CrossRefGoogle Scholar

† As stated in the Introduction this section is due to Kaufmann and myself jointly.