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# Domains.

Definition 4.1   Let A be a point and let be a given number. The open ball whose center is A and whose radius is is the set .

A subset V of the plane (resp. of the 3-dimensional space) is open if, for every point , there exists an such that .

The complementary of an open subset is called a closed subset.

The set of all the points P in the plane (resp. in the space) such that every open ball centerd at P contains both interior points of a subset D and points exterior to D, is called the frontier of D.

Example 4.2

• The whole plane is open.
• The unit disk is open. For any , denote ; then .
• The closed unit disk'' is not open: for every point P on the unit circle, an open ball whose center is at P contains points of the exterior of V (cf Fig  1).
• The unit circle is the frontier of the unit disk.

Next: Level curves. Up: Multivariable functions. Previous: Multivariable functions.
Noah Dana-Picard
2001-05-30