Colloquium publications.

ANALYSIS SITUS.

5

transformation of the curve. The Heine-Borel theorem is another case in point.
The theorems of Analysis Situs may be divided somewhat
roughly into two classes, those dealing essentially with continuity
considerations (of which the theorem on perfect sets of points
cited above may serve as an illustration), and those having an
essentially combinatorial character. It is the theorems of the
latter class which will occupy most of our attention in the following pages, though we shall continually make use of theorems of
the former class without proving them.
Singular Complexes
7. Let F be a correspondence between a O-dimensional complex Co and a set of points [P] of any complex C (for the present,
C is 0- or 1-dimensional) in which each point of Co corresponds
to a single P and each P is the correspondent of one or more
points of Co. The object obtained by associating any point X
of Co with the point P which is its image under F will be denoted
by F(X) and called a point on C; it is said to coincide with P
and P to coincide with it. The set of all points F(X) on C
is called a O-dimensional complex on C. If any P is the correspondent of more than one point X of Co, F(X) is called a singular point and the complex on C is said to be singular.
8. Let C1 be a generalized one-dimensional complex and let
F be a continuous correspondence between C1 and a set of points
[P] of a complex C, in which each point of C1 corresponds to a
single P and each P is the correspondent of at least one point of
C1. The object obtained by associating any point X of C1
with the point P which is its image under this correspondence
will be called a point on C and is uniquely denoted by the functional notation 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 C is in a (1-1)
continuous correspondence with the points of C1 and thus constitutes a one-dimensional complex C1' identical in structure with
C1. The one-dimensional complex C1' is said to be on C. If any
of the points P is the correspondent under F of more than one