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ANALYSIS SITUS.

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constitute a linear graph for which ao = 4 and ac = 6. A linear
graph is not necessarily assumed to lie in any space, being defined
in a purely abstract way. It is obvious, however, that if ao
points be chosen arbitrarily in a Euclidean three-space they can
be joined by pairs in any manner whatever by a1 non-intersecting
simple arcs. Therefore, any linear graph may be thought of as
situated in a Euclidean three-space.
For some purposes it is desirable to use the term one-dimensional complex to denote a more general set of 1-cells and 0-cells
than that described above. For example, a 1-cell and its two
ends form a one-dimensional complex according to the definition
above, but a 1-cell by itself or a 1-cell and one of its ends do not.
In the following pages we shall occasionally refer to an arbitrary
subset of the 1-cells and 0-cells of a linear graph as a generalized
one-dimensional complex.
4. A transformation F of a set of points [X] of a complex C1
into a set of points [X'] of the same or another complex is said
to be continuous if and only if it is continuous in the sense of ~ 2
on each complex composed of a 1-cell of C1 and its ends (i.e.,
if the transformation effected by F on those Xs which are on such
a 1-cell and its ends is continuous). A (1-1) continuous transformation of a complex into itself or another complex is called,
following Poincare, a homeomorphism. Two complexes related
by a homeomorphism are said to be homeomorphic.
The set of all homeomorphisms by which a linear graph is
carried into itself obviously forms a group. Any theorem about
a linear graph which states a property which is left invariant by
all transformations of this group is a theorem of one-dimensional
Analysis Situs. The group of homeomorphisms of a linear graph
is its Analysis Situs group.
Order Relations on Curves
5. By an open curve is meant the set of all points of a complex
composed of a 1-cell and its two ends. By a closed curve is meant
the set of all points of a complex C1 consisting of two distinct
0-cells a0~, a2~ and two 1-cells a1l, a21, each of which has a~1