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ANALYSIS SITUS.

53

each point of C' corresponds to one point of a set of points [P]
of C2 while each P is the correspondent of one or more points
of C'. If C' is of one or two dimensions we require F to be continuous. Under these conditions, any point X of C' associated
with the P to which it corresponds under F is called a point on C2;
it is referred to as the image of X under F and is uniquely denoted
by F(X); it is said to coincide with P and P is said to coincide
with it. The set of all points F(X) on C2 is in a (1 - 1) continuous correspondence with the points of C' and thus constitutes
a complex C" identical in structure with C'. The complex C"
is said to be on C2. If any of the points P is the correspondent
under F of more than one point of C', C" is called a singular
complex on C2 and the point P in question is called a singular
point. If F is (1 - 1), C" is said to be non-singular. A cell of C"
is said to coincide with a cell of C2 if and only if (1) each point of
the cell of C" coincides with a point of the cell of C2 and (2) the
correspondence thus set up is (1 - 1).
In case C" is two-dimensional and such that there is at least
one point of C" on each point of C2 and if, furthermore, there
exists for every point of C" a neighborhood which is a nonsingular complex on C2, then C" is said to cover C2. In case the
number of points of C" on each point of C2 is finite and equal
to n, C" is said to cover C2 n times (cf. ~ 9, Chap. I).
34. Any 2-circuit which is not a manifold can be regarded as a
singular manifold. For let C2 be an arbitrary 2-circuit. Each
of its edges, ail, is incident with an even number, 2ni of 2-cells.
These 2-cells may be grouped arbitrarily in ni pairs no two of
which have a 2-cell in common; let these be called the pairs of
2-cells associated* with ail. Let C2' be a 2-circuit on C2 such that
(1) there is one and but one 2-cell of C2' coinciding with each
2-cell of C2, (2) there are ni 1-cells of C2' coinciding with each
1-cell ail of C2, each of the ni 1-cells being incident with a pair
of 2-cells of C2' which coincide with one of the pairs of 2-cells
associated with ail, and (3) there is one 0-cell of C2' coincident
with each O-cell aio of C2, this O-cell being incident with all the
*Cf. ~ 22, Chap. I.