San José State University |
---|
applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
---|
and Its Extensions |
The Jordan Curve Theorem says that a simple (non-selfintersecting) closed curve in a plane partitions the plane into an inside region and outside region. The two regions are internally connected but an outside point cannot be connected with an inside point without cross the curve.
Note that this is not true for a curve in 3 dimensional space. Instead it is a simple closed surface that partitions the space into an inside region and an outside region.
Camille Jordan proposed this theorem and published a proof for it in 1887. Later some mathematicians claimed Jordan's proof was not valid. Oswald Veblen is given credit for providing a rigorous proof of the theorem. More recently mathematicians investigating Jordan's proof found it not to be flawed.
Consider a plane with a polar coordinate system; i.e., a point is given by its polar coordinates (r, θ). A curve is a continuous mapping of a unit interval [0, 1] into that plane. That is to say a curve is given by a pair of continuous functions {r(t), θ(t): 0≤t≤1 }.
Any closed curve in the plane is topologically equivalent to a unit circle centered on the origin. A unit circle, as a curve, is given by radius and angle functions of the form
Any point (R, Θ) is inside the circle if R<1 and outside the circle if R>1.
A region is connected if, for any two points in the region, there is a curve that goes from one point to the other and lies entirely within the region. Let (R_{0}, Θ_{0}) and (R_{1}, Θ_{1}) be any two points inside the circle. One curve connecting them starts at (R_{0}, Θ_{0}) and keeps the same radius and changes the angle from Θ_{0} to Θ_{1}). It then maintains the same angle Θ_{1} and changes the radius from R_{0} to R_{1}, as indicated below.
The explicit functions for the curve can be presented but there does not seem to be any reason to do so.
Since the radius coordinate for this curve is never greater than the larger of R_{0} and R_{1} the points of the curve are always inside of the circle.
Likewise for any two points outside the circle a similar sort of curve can be constructed, as shown below.
Let (r(t), θ(t)) be any curve starting at an inside point (R_{0}, Θ_{0}) and ending at an outside point (R_{1},Θ_{1}). Since R_{0}<1 and R_{1}>1, by the Intermediate Value Theorem, there exists some t_{c} such that r(t_{c}) is equal to 1. Thus the curve has to intersect the circle.
The spherical coordinate system for 3 space is of the form (r, θ, φ) where r can take on any nonnegative value, θ ranges from 0 to 2π and φ from 0 to π.
Any simple (non-selfintersecting) closed surface is topologically equivalent to a unit sphere. A unit sphere partitions 3 Space into inside points and outside points. A point (R, Θ, Φ) is an inside point if R<1 and an outside point if R>1.
Any two inside points may be connected by a curve in which the r and θ are held constant while φ changes from Φ_{0} to Φ_{1}. Then r and φ are held constant while θ changes for Θ_{0} to Θ_{1}. And finally θ and φ are held constant while r changes for R_{0} R_{1}. Thus all inside points are connected. Similarly any two outside points are connected. Finally, as in the case of a plane, any curve connecting an inside point with an outside must cross the unit sphere.
N Space has a coordinate system consisting of a radius and (n-1) angle variables. Any simple closed (n-1) dimensional surface in n space is topologically equivalent to a unit (n-1) sphere. The n dimesional space is partitioned by the (n-1) sphere into inside points (r<1) and outside points r>1. The inside points are connected, the outside points are connected but an inside point cannot be connected to an outside point without passing through the (n-1) sphere.
A curve with one point of intersection is topologically equivalent to two unit circles which are tangent at one point. Such a self-intersecting curve partitions the plane into two regions interior to the curve and the region outside of the curve. Points in the two regions interior to the curve cannot be connected by a curve without that curve having to pass through points on the curve.
(To be continued.)
HOME PAGE OF Thayer Watkins |