Definition: Closed Curve, Open Curve

A curve \(f:I\to\mathbb R^n.\) is called closed if the interval \(I\) is closed, i.e. \(I=[a,b]\) and if \(f(a)=f(b)\).

A closed curve \(f:[a,b]\to\mathbb R^n\), whose restriction to the open real interval \((a,b)\), i.e. the function \({f|}_{(a,b)} : I \to \mathbb R^n\), is a simple curve, is called a simple closed curve.

Examples of closed curves in the plane \(\mathbb R^2\)

The closed curve to the left is a simple closed curve, while the curve right to that curve is a closed curve, but not a simple closed curve.


(Image Source: bookofproofs)

Examples of open curves in the plane \(\mathbb R^2\)


(Image Source: bookofproofs)

Definitions: 1 2 3

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  1. Matoušek, J; Nešetşil, J: "Invitation to Discrete Mathematics", Oxford University Press, 1998