1550251515-Classical_Complex_Analysis__Gonzalez_

(jair2018) #1
Topology of Plane Sets of Points

Example Consider in ( C, d) 'the set


A= {z: JzJ < 1} U {z: z^2 = -1}


Then

Int 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.



  1. A is open iff A = Int A.

  2. A is closed iff A = A.


4. AC B implies that IntA C IntB and AC B.


  1. CIA = CIA'.


6. CIA= An A'.


2.8 ISOLATED POINTS, LIMIT POINTS, AND

CONTACT POINTS. DERIVED SET. DENSE AND

PERFECT SETS

97

Definition 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.
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