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Domains.

Definition 4.1   Let A be a point and let $\varepsilon >0$ be a given number. The open ball whose center is A and whose radius is $\varepsilon$ is the set $B(A,\varepsilon)= \{ P \vert AP< \varepsilon \}$.

A subset V of the plane (resp. of the 3-dimensional space) is open if, for every point $P \in V$, there exists an $\varepsilon >0$ such that $ B(A,\varepsilon) \subset V$.

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       


  
Figure 1: Open balls whose center is on the unit circle.
\begin{figure}
\mbox{\epsfig{file=unitdisk.eps,height=4cm} }
\end{figure}


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