Colloquium publications.

ANALYSIS SITUS.

95

proved that Ki can be regularly subdivided into a complex K2
such that for each j-cell of Ki there is a star of cells of On to
which it is interior. A correspondence A is now defined as a
correspondence between the vertices of Ki and those of Cn by
which each vertex of Ki which is interior to a star of cells of Cn
having a vertex of Cn as center corresponds to that vertex of Cn,
and by which each vertex of Ki which is on the boundary of two
or more stars of cells of Cn having vertices of Cn as centers
corresponds to one of these vertices of Cn.
Since every point of Cn is on or on the boundary of some star
of cells of Cn with center at a vertex of Cn, a correspondence A
determines a unique vertex of Cn for each vertex of Ki. Moreover since any cell of Ki is on a star of cells of Cn its vertices
correspond to vertices of a single cell of Cn. Hence the correspondence A makes each cell of Ki correspond to a cell of Cn
of the same or lower dimensionality.
37. Let the r-cells of Cn be denoted by cfj (r = 0, 1, 2, * * *, n;
j= 1, 2,... ar) and those of Ki by kr (r= 0, 1, 2,..., i;
j = 1, 2, * * * r). Each 0-cell kj~ of Ki can be joined to the 0-cell
of C, to which it corresponds under the correspondence A by a
straight 1-cell bj1; or, if kjO coincides with the point to which it
corresponds, by a singular 1-cell bj1 coinciding with kj~. Similarly, for each 1-cell kj1 of Ki, a 2-cell bj2 can be constructed by
joining each point of kj1 to a point of the corresponding cell of Cn
by a 1-cell which is either straight or coincident with a point.
By a similar construction there is determined for every cell kjr
of Ki a cell br+1 composed of 1-cells joining points of kjr to points
of the cell of Cn to which kjr corresponds under the correspondence
A. The (i + 1) dimensional complex composed of the cells
bj+1l and their boundaries is denoted by B2+,. It is such that
the incidence relations of br+l and bqr are the same as those of
kpr and kqr-1.
38. If Ki is an i-circuit, all i-cells bji (j = 1, 2,...,,i-x)
must cancel out when the boundaries of the (i + 1)-cells bji+l
(j = 1, 2,..., Afi) are added together (mod. 2). Hence the
boundary of Bi+1 consists either of Ki alone or of Ki and a set
of i-circuits K/' composed of cells of Cn. That is to say