Curve - Topology

Topology

In topology, a curve is defined as follows. Let be an interval of real numbers (i.e. a non-empty connected subset of ). Then a curve is a continuous mapping, where is a topological space.

• The curve is said to be simple, or a Jordan arc, if it is injective, i.e. if for all, in, we have implies . If is a closed bounded interval, we also allow the possibility (this convention makes it possible to talk about "closed" simple curves, see below).

In other words this curve "does not cross itself and has no missing points".

• If for some (other than the extremities of ), then is called a double (or multiple) point of the curve.

• A curve is said to be closed or a loop if and if . A closed curve is thus a continuous mapping of the circle ; a simple closed curve is also called a Jordan curve. The Jordan curve theorem states that such curves divide the plane into an "interior" and an "exterior".

A plane curve is a curve for which X is the Euclidean plane—these are the examples first encountered—or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). However, in the case of algebraic curves it is very common not to restrict the curve to having points only defined over the real numbers.

This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, without thickness and drawn without interruption, although it also includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover a square in the plane (space-filling curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.