Jordan curve theorem

The Jordan curve theorem implies that this procedure is well defined.

This observation may be mathematically proved using the Jordan curve theorem.

In dimension two this is the Jordan curve theorem.)

For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way.

The Jordan curve theorem states that such curves divide the plane into an "interior" and an "exterior".

Page 7 introduces the Jordan curve theorem.

The statement of the Jordan curve theorem may seem obvious at first, but it is a rather difficult theorem to prove.

Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to implicit use of the Jordan curve theorem.

By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.

Oswald Veblen proves the Jordan curve theorem.

The prototype here is the Jordan curve theorem, which topologically concerns the complement of a circle in the Riemann sphere.

The Jordan curve theorem is named after the mathematician Camille Jordan, who found its first proof.

A short elementary proof of the Jordan curve theorem was presented by A. F. Filippov in 1950.

For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem.

The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem.

Crucial to this definition is the fact that every simple closed curve admits a well-defined interior; that follows from the Jordan curve theorem.

Some new elementary proofs of the Jordan curve theorem, as well as simplifications of the earlier proofs, continue to be carried out.

A converse to the Jordan curve theorem, proved by Schönflies, states that a subset of the 2-sphere is a simple closed curve if it:

He studied the foundations of geometry with Hilbert at Göttingen in 1899, and obtained a proof of the Jordan curve theorem for polygons.

It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan curve theorem.

In the context of this work, Brouwer also generalized the Jordan curve theorem to arbitrary dimension and established the properties connected with the degree of a continuous mapping.

Applications of the fundamental groupoid on a set of base points to the Jordan curve theorem, Covering spaces, and orbit spaces are given in Ronald Brown's book.

Due to the importance of the Jordan curve theorem in low-dimensional topology and complex analysis, it received much attention from prominent mathematicians of the first half of the 20th century.

In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies.

Some examples are the Hahn-Banach theorem, König's lemma, Brouwer fixed point theorem, Gödel's completeness theorem and Jordan curve theorem.