Colloquium publications.

ANALYSIS SITUS.

83

go beyond the combinatorial properties of C, and make use of
the geometrical properties of the cells.
Theory of the n-cell
17. The combinatorial properties of a complex Cn which
have been discussed above have an elementary application in the
theory of the subdivision of a Euclidean space by generalized
polyhedra. A system of (n - 1)-spaces in an n-space subdivide
the n-space into a set of n-dimensional convex regions. They
intersect in a number of (n - 2)-spaces which subdivide each
(n - 1)-space into a set of (n - 1)-dimensional convex regions
which bound the n-dimensional convex regions. The (n - 2)spaces have (n - 3)-spaces in common which divide the (n - 2)spaces into convex regions, and so on. Thus the set of (n - 1)spaces defines a subdivision of the n-space into a set of cells which
can be treated by the methods described above. Any k-circuit
formed from the k-dimensional convex regions is a generalized
polyhedron. Any such k-circuit' bounds a (k+1)-dimensional
complex composed of convex (k+l)-cells.
A treatment of the theory of polyhedra from this point of
view by the author is to be found in the Transactions of the
American Math. Soc., Vol. 14 (1913), p. 65. (See also the
correction Vol. 15, p. 506.) Earlier and later treatments without
the machinery used here are to be found in the papers by N. J.
Lennes, Am. Journ. of Math., Vol. 33 (1911), p. 37, and Lilly
Hahn, Monatshefte fur Math. u. Phys., Vol. 25 (1914), p. 303.
Since an n-cell is homeomorphic with a Euclidean space all this
is the most elementary part of the theory of the n-cell.
18. As in ~ 8, Chap. II, we can define a system of curves in
any n-cell ail (i = 1, 2,.* *, cn) which have the properties of the
system of straight lines interior to a simplex in a Euclidean space.
It is only necessary to set up a (1-1) continuous correspondence
Fi between the interior and boundary of the n-cell and the
interior and boundary of a simplex and to regard as straight
those curves in the n-cell which are images of straight lines in the
simplex.