Colloquium publications.

4

THE CAMBRIDGE COLLOQUIUM.

and a2~ as ends but which have no common points (fig. 2).' The
most elementary theorems about curves are those which codify
the order relations. They may be stated (without proof) as
follows:
Let us denote a 1-cell and its ends by a1, ai~ and a20. If a3~
is any point of a1, there are two 1-cells al, and a2' such that a,'
has ai~ and a3~ as it ends, a2' has a3~ and a2~ as its ends, and every
point of a' is either on all or a21 or identical with a3~. The 1-cell
al is said to be separated into the 1-cells all and a2l by the 0-cell a3~.
al
FIG. 2
FIG. 2.
A O-cell is said to be incident with a 1-cell if and only if it is
an end of the 1-cell; and under the same conditions the 1-cell
is said to be incident with the 0-cell. It follows directly from
the theorem on separation in the paragraph above that n distinct
points of the 1-cell a' determine n + 1 1-cells such that the n
points (or 0-cells) may be denoted by b1~, b2~, *., bn~ and the
n + 1 1-cells by bl', b21,.* *, bn+1 in such a way that each cell
is incident with the cell which directly precedes or directly
follows it in the sequence a1~, bl,, b1~, b21,.., b,*, bn,+lla2.
If b1~, b2~, **., bn~ are n distinct points of a closed curve the
remaining points of the curve constitute n 1-cells bi' (i = 1, 2,
* *, n), no two of which have a point in common, such that each
bi~ is incident with just two of them.
6. A little reflection will convince the reader that many
of the theorems about functions of one real variable and
about linear sets of points belong to one-dimensional Analysis
Situs. As an example we may cite the theorem that any
nowhere dense perfect set of points on a closed curve can be
transformed into any other such set by a (1-1) continuous