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Note on Topics of regions separated by boundaries

Published online by Cambridge University Press:  20 January 2009

R. F. Muirhead
R. F. Muirhead
Affiliation:
Glasgow.
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Sr denotes an infinite space of r dimensions. Such a space is divided into two regions by a Sr−1. An infinite line is divided into n + 1 regions by n points. An infinite plane or other surface topically equivalent to it is divided into two regions by an infinite line.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , February 1924 , pp. 111 - 114
Copyright
Copyright © Edinburgh Mathematical Society 1924

References

* Cayley, (Mess. Math., IV, p. 167)Google Scholar gives this formula in the form ½(n 2+n+2), and the corresponding formula for φ3(n), ½(n 2+5n + 6).

* A particular case of this Theorem, viz., φn(n + l) = 2n + 1 − 1 is given in Schoute's, Mehrdimensionale Geometric, II (Leipzig 1905), 6.Google Scholar