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2008-05-18T20:09:17Z

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Vipul: New page: ==Definition== A '''simple closed curve in the plane''' or '''simple loop in the plane''' is a simple closed curve in the Euclidean plane. In other words, it is a subset of $...$

2008-04-04T19:54:46Z

New page: ==Definition== A '''simple closed curve in the plane''' or '''simple loop in the plane''' is a simple closed curve in the Euclidean plane. In other words, it is a subset of <math>...

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==Definition==

A '''simple closed curve in the plane''' or '''simple loop in the plane''' is a [[simple closed curve]] in the [[Euclidean plane]]. In other words, it is a subset of <math>\R^2</math> that satisfies the following equivalent conditions:

* It can be obtained as the image of <math>S^1</math> under an injective continuous map
* It is homeomorphic to <math>S^1</math>

Note that these definitions are equivalent since <math>S^1</math> is compact, the plane is Hausdorff, and [[tps:injection from compact to Hausdorff implies embedding|any injective map from a compact space to a Hausdorff space is an embedding]].

==Facts==

By the [[Jordan curve theorem]], the simple closed curve divides its complement into exactly two components. One of the components is bounded, and is termed the ''interior'' of the curve, while the other component is unbounded and is termed the ''exterior'' of the curve.

Simple closed curve in the plane - Revision history

Vipul: 1 revision

Vipul: New page: ==Definition== A '''simple closed curve in the plane''' or '''simple loop in the plane''' is a simple closed curve in the Euclidean plane. In other words, it is a subset of ...

Vipul: New page: ==Definition== A '''simple closed curve in the plane''' or '''simple loop in the plane''' is a simple closed curve in the Euclidean plane. In other words, it is a subset of $...$