Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T07:35:27.739Z Has data issue: false hasContentIssue false
  • English
  • Français

The Cantor Manifolds Lying on a Closed Surface. Part II

Published online by Cambridge University Press:  24 October 2008

H. D. Ursell
Get access Rights & Permissions [Opens in a new window]

Extract

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.