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ANALYSIS SITUS.

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decomposed into n 2-cells a12, a22, *.., an2 such that the sum
of their boundaries (mod. 2) is the boundary of a2 and such
that the incidence relations between them and a1l, a21, a, an
are the same as the incidence relations between the 0-cells and
1-cells of a 1-circuit.
Conversely, if all, a21, *, an1 and a12, a22, *., a,2 are 1-cells
and 2-cells all incident with the same point a~ and also incident
with one another in such a way that the incidence relations
between the 1-cells and 2-cells are the same as those between the
0-cells and 1-cells of a 1-circuit, then the point a~ and the points
of a1, a21, *, an and a12, a22, *, an2 constitute a 2-cell a2
which is bounded by the sum (mod. 2) of the boundaries of the
2-cells a12, a22, *, an2.
10. The first of the theorems in the last section is a special
case of the theorem that any 1-cell which is in a 2-cell and joins
two points of its boundary decomposes the 2-cell into two 2-cells.
This more general theorem depends on the theorem of Jordan,
that any simple closed curve in a Euclidean plane separates the
plane into two regions, the interior and the exterior; and also on
the theorem of Schoenflies that the interior of a simple closed
curve is a 2-cell of which the curve is the boundary.
We shall not need to use these more general forms of the
separation theorems because we need, in general, merely the
existence of curves which separate cells, and this is provided for
in the theorems of the last section. In connection with the
Jordan theorem, reference may be made to the proof by J. W.
Alexander, Annals of Math., Vol. 21 (1920), p. 180.
Maps
11. With the aid of the theorems on separation a 2-cell a2 may
be subdivided into further 2-cells as follows: Let any two points
a1~ and a2~ of the boundary of the 2-cell be joined by a straight
1-cell a1l consisting entirely of points of the 2-cell. The 2-cell
is thus separated into two 2-cells a12 and a22. The boundary
of a2 is likewise separated into two 1-cells all and a2' which have a1~
and a2~ as ends. The 0-cells, 1-cells and 2-cells into which a2 is