Topology of Plane Sets of Points 85To see clearly why we distinguish between the mapping (which we call
the oriented closed segment) and its graph (which we may call the closed
segment), consider the mapping(2.4-14)
defining the oriented closed segment [z 2 , z 1 ]. The mappings defined
by (2.4-12) and (2.4-14) are different since they induce a different ordering
of the points of the geometric segment. However, they both have the same
graph, the set
[z2,z 1 ]* = {z: z = (1-t)z 2 + tz 1 , 0 < t < 1}
being the same set defined in (2.4-13).
- Oriented open segment. The oriented open segment, denoted (z 1 ,z 2 ),
is defined by the mapping
z = (l-t)z1 +tzz, O<t<l (2.4-15)
and its graph is indicated by (z1, zz).
The oriented half-open segments ( z 1 , z 2 ] and [ z1, zz) are defined in a
similar fashion, and the corresponding graphs are denoted by (z 1 , z 2 ] and
[ z1, Zz) *, respectively. ·
- Polygonal line. A polygonal line in C is the set
n
P = LJ [zk-1,zk]*
k=l
where n :'.:'.: 2. It is said that the polygonal line joins the points z 0 and Zn·
Each set [zk-l, Zk]* is called a segment or a side of the polygonal line. if
zo =Zn, the polygonal line is said to be closed (sometimes called a polygon).
- Convex set. A nonempty set J( C C (more generally, in any vector
space) is said to be convex if for any two points z 1 , z 2 E K, we have
[z1,z2]* C K.
Examples C itself, a half-plane, and an open disk, are convex sets.
- Starlike set. A nonempty set A C C is starlike if there is some point
z 0 C A such that for any z E A, we have [z 0 , z]* C A. The point zo is
called the star center of A. Obviously, any convex open set is starlike, and
any point of the set can be chosen as a star center. There are starlike s~ts
that are not _convex: for instance, a five-pointed star. - Half-planes. An oriented line L: z = a + bt, -oo < t < +oo
determines a right half-plane P 1 and a left half-plane Pz (Fig. 2.4). The
right half-plane P 1 can be characterized analytically as a set of points z
such that
z-a
Im --< 0
b(2.4-16)