# Jordan Curve Definition

The Jordan curve theorem states that the set complement in a plane of a Jordan curve consists of two connected components that is the curve divides the plane in two non-intersecting regions that are both connected. If is a simple closed curve in then the Jordan curve theorem also called the Jordan-Brouwer theorem Spanier 1966 states that has two components an inside and outside with the boundary of each.

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### In topology a Jordan curve sometimes called a plane simple closed curve is a non-self-intersecting continuous loop in the plane.

Jordan curve definition. THE CASE OF POLYGONS. Definition of Jordan curve theorem. Anything with a round or oval shape formed by a curve that is closed and does not intersect itself.

Also HomeomorphismNamed after C. We have proved the Jordan curve theorem under the third. Theorem 21 A conformal map φ of a Jordan domain F onto a Jordan domain G can be extended to a homeomorphism of F onto G.

It bounds two Jordan domains. A continuous simple closed curve or Jordan curve in a topological space or convergence space locale etc X X is the image of a continuous injection to X X from the unit circle S 1 S1. Jordan Curve Theorem.

Princetons WordNet 000 0 votes Rate this definition. I sort of understand what its trying to say. Also called a simple closed curve.

A Jordan curve is said to be a Jordan polygon if C can be covered by finitely many arcs on each of which y has the form. Cases can not happe ton a Jordan curve. Sorry for the trivial question.

It is not known if every Jordan curve contains all four polygon vertices of some square but it has been proven true for sufficiently smooth curves and closed convex curves Schnirelman. Closed curve – a curve such as a circle having no endpoints. This map itself is a continuous parametrization.

Also is the Jordan Curve Theorem just saying that a closed loop partitions the plane into an inside and outside. Every simple closed curve divides the plane into two regions for which it is the common boundary First Known Use of Jordan curve theorem. The Jordan curve theorem says that a simply closed curve separates a simply connected surface into two components.

Simple closed curve Jordan curve noun. Jordan curve plural Jordan curves topology A non-self-intersecting continuous loop in the plane. A fundamental theorem of topology.

See also Line curve. Theoretically these three definition shall be equivalent. Suppose f is a piece-wise circular simple closed curve and is a piece-wise circular arc.

Stated differently the Jordan curve is a continuous image of the segment. Suppose we have the. Based on the definition of discrete surfaces we give three reasonable definitions of simply connected spaces.

A simple closed curve. Of the curve The word continuous is generally assumed so that one speaks simply of a. Loop – anything with a round or oval shape formed by a curve that is closed and does not intersect itself.

The Jordan curve theorem is a standard result in algebraic topology with a rich history. A Jordan curve Γ is said to be a Jordan polygon if there is a partition Θθθ θ 01 n of the interval 02π ie 02. By Jordan curve we mean the homeomorphic image of T.

Prelhninary Lemmas We had defined when an arc is said to cross a circle. Ycost sint Xt fi pt a with constants H pa. Jordan curve – a closed curve that does not intersect itself.

A plane simple closed curve is also called a Jordan curveIt is also defined as a non-self-intersecting continuous loop in the plane. The proof first proves the Jordan curve theorem for polygons and then uses an approximation argument to derive the Jordan curve theorem in general. A homeomorphic image of a circle cf.

Thu Fs is a closed polygon without self intersections. Jordan who suggested the definition. The Jordan curve holds theorem for every Jordan polygon f.

A Jordan curve is a plane curve which is topologically equivalent to a homeomorphic image of the unit circle ie it is simple and closed. Jordan curve theorem definition the theorem that the complement of a simple closed curve can be expressed as the union of two disjoint sets each having as. Jordan Curve Similarly a closed Jordan curve is an image of the unit circle under a similar mapping and an unbounded Jordan curve is an image of the open unit interval or of the entire real line that separates the plane.

Though Im not sure what continuous image means. To ease our way into the formal proof let us recall Thomassens argument showing that if C is a simple closed polygonal curve in the. A closed curve that does not intersect itself.

A closed curve is similar to a circle. A curve is similar to a line segment. We broaden the definition of crossing as follows.

1950 Joseph Leonard Walsh The Location of Critical Points of Analytic and Harmonic Functions American Mathematical Society page 242 When μ is small and positive the locus 1 consists of a Jordan curve near each of. All the pieces to conclude the proof of the Jordan Curve theorem. The geometric locus of points M x y of a plane the coordinates of which satisfy the equations x ϕ t and y ψ t where ϕ and ψ are continuous functions of the argument t on some segment a b.

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