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## 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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### Bibliography (further reading)

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