Jordan curve theorem
It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Hales wrote:. Once you can prove that the curve has a regular neighborhood which is a normal bundle, then Stillwell's proof should still work.
King of spades: Eskenazis, A. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice.
Logos Verlag Berlin GmbH. Namespaces Article Talk. But there are two observations one can make which are independent of that.
Dalang , Mountford : Nondifferentiability of curves on the Brownian sheet
The same fractal as above, magnified fold, where the Mandelbrot set fine detail resembles the detail at low magnification. Uh Oh. Sign me up! Hot Network Questions. Ryan Budney Ryan Budney In particular, for any point P in the interior region and a point A on the Jordan curve, there exists a Jordan arc connecting P with A and, with the exception of the endpoint A , completely lying in the interior region.
Nowhere differentiable functions of analytic type on products of finitely connected planar domains
You have access to this content. Complex analysis — Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. Jordan curve theorem mathematics.
We call such curves rectifiable. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Please try again later. At the bottom of the article, feel free to list any sources that support your changes, so that we can fully understand their context.
Note that assuming that the parameterisation is only continuous is not sufficient, since there are curves which are continuous everywhere but nowhere differentiable and which have infinite length see the Koch curve. The Jordan Curve theorem is actually pretty easy to prove if you assume the curve is smooth or piecewise linear. Let be an interval with and let be two continuous functions such that for all The set. This is proved by induction in k using the Mayer—Vietoris sequence. Email Address never made public.
This game can be made more complicated by there being more than one disjoint simple closed curve in the diagram. Regions which are both -simple and -simple are in the following called simple.
Yes that is a requirement for the JCT. Let now be a continuous vector field and let be rectifiable. Let C be a Jordan curve in the plane R 2.