**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)

| | | | | Contributors: *bookofproofs* | References: [1209]

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[1209] **Matoušek, J; Nešetřil, J**: “Invitation to Discrete Mathematics”, Oxford University Press, 1998

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