A curve has already been defined in Ch. IX. The definition is repeated here.

DEF. A plane set of points, dense nowhere in the plane, such that, given any small norm e, and describing round each point of the set a small region of span less than e, these small regions generate a single region Re, whose span does not decrease indefinitely, is called a curved arc, or shortly a curve.

The following then follow from the investigations on regions:—

A curve is never a point and never a region, and, only when the span of the region Re in one direction diminishes without limit as e does so, is it a stretch (segment of a, straight line).

The points of a curve form a connected set.

A closed connected set dense nowhere in the plane is a curve, and is said to be a complete curve.

The points of a curve may or may not form a closed set: the non-included limiting points may be finite, or countably infinite, or more than countable.

DEF. An arc, every one of whose points is a point of a certain curve, is called an arc of that curve.

The following property of a curve is an immediate consequence of the definition :—

Given any two points P and Q of a curve, there is at least one arc of the curve PQ not containing P nor Q, but having both these points as limiting points.