Topology of Plane Sets of PointsExample Consider in ( C, d) 'the set
A= {z: JzJ < 1} U {z: z^2 = -1}
ThenInt A= {z: Jzl < 1},
Ext A= {z: lzJ > 1},
A= {z: Jzl ~ 1}
CIA= {z: lzl = 1}
Theorem 2.4 The following properties hold:1. Int A c A c A.
- A is open iff A = Int A.
- A is closed iff A = A.
4. AC B implies that IntA C IntB and AC B.
- CIA = CIA'.
6. CIA= An A'.
2.8 ISOLATED POINTS, LIMIT POINTS, AND
CONTACT POINTS. DERIVED SET. DENSE AND
PERFECT SETS
97Definition 2.18 Let (S, d) be a metric space and let A C S. A point
a E A is said to be an isolated point of A iff {a} = An N5(a) for some
8 > 0, i.e., iff the intersection of A with some neighborhood of a is the
singleton {a}.Definition 2.19 A point b E S (which may or may not belong to A) is
a limit point or an accumulation point of A C S iff A n NHb) f. 0 for
every 8 > 0. Then it follows that every N5(b) with 8 > 0 contains infinitely
many points of A. Alternatively, b is a limit point of A iff b E A and b
is not an isolated point of A.
Definition 2.20 A point c E S is called a contact point, an adherent
point, or a point of closure of the set A C S iff c E A. It is clear that every
contact point of A is either an isolated point of A or a limit point of A.Example Consider in ( C, d) the set
A={z: Rez>O}U{-1,-2,-3}
The points -1, -2, and -3 are isolated points of A, while all points in
{z: Rez ~ 0} are the limit points of A. All the points in
A={z: Rez~O}U{-1,-2,-3}
are the contact points of A.