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The open unit disc around P (where P is a given point on the plane), is the set of points whose distance from P is less than one:

${\displaystyle D_{1}(P)=\{Q:\vert P-Q\vert <1\}}$.

A closed unit disc is the set of points whose distance from P is less than or equal to one:

${\displaystyle {\bar {D}}_{1}(P)=\{Q:|P-Q|\leq 1\}}$.

Unit discs are a special case of unit balls. Without further specifications, the term unit disc is used for the open unit disc about the origin, ${\displaystyle D_{1}(0)}$.

The specific set of points in the unit disc, and hence, its visual appearance, depends on the metric used. See the diagram for examples.

With the standard metric, unit discs look like circles of radius one, but changing the metric changes also the corresponding set of points and therefore the shape of the discs.

For instance, with the taxicab metric discs look like squares, while on the Chebyshev metric a disc is shaped like a rhombus (even though the underlying topologies are the same as the Euclidean one). In the Euclidean plane, the area of the unit disc is π, but with the Chebyshev metric the area of unit disk is 4.

See also

• open interval
• closed interval
• open set
• closed set
• ball (mathematics)
• unit circle

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