
Section 1.3 Screens and roundings 29
To illustrate the definition and the theorems given in this chapter, let us consider a
few examples.
Examples. 1. Let Z be a bounded set of complex numbers Z :Df :D C i 2
C jjjr ^jjrg. The power set fPZ, g is a complete lattice. We consider
the set IZ of all rectangles of PZ with sides parallel to the axes. The elements of IZ
are intervals in the complex plane. Also let ¿ 2 IZ. We show that fIZ, g is an upper
screen of fPZ, g.
To see this, by Theorem 1.15 we have only to show that for every subset A IZ,
the intersection, which is the infimum in fPZ, g, is an element of IZ.IfA D ¿,we
have
inf
PZ
¿ D inf
IZ