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*.

- 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.