1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 77

each discussion the master or universal set is denoted by U. In what follows

we shall have U = C, or U = C*, in most cases.

Definition 2.1 If there exists a one-to-one mapping of A onto B, it is

said that A and B can be put into one-to-one correspondence (briefly, 1-1

correspondence), or that the sets A and B are equivalent, or that they have

the same cardinal number. We express that the sets A and B are equivalent

by writing A ,...., B. This relation has the following properties:

1. Reflexive: A ,...., A


2. Symmetric: if A ,...., B, then B ,...., A.


3. Transitive: if A ,...., B and B ,...., C, then A ,...., C.
   In general, any relation having those three properties is said to be an
   equivalence relation.
   Examples


4. Any two segments (considered as sets of points) are equivalent.

2. The set J = { 1, 2, ... , n, ... } of the positive integers is equivalent to

the set E = {2, 4, ... , 2n, ... } of the even positive integers.

3. The set Q of the rational numbers is equivalent to J.


4. The set JR of the real numbers is not equivalent to J (Cantor).
   Definitions 2.2 Let Jn = { 1, 2,. .. , n}. Then given a set A, we have:


5. A is finite if either A = 0 or A ,...., Jn for some n.


6. A is infinite if A is not finite. In a positive characterization A is infinite
   iff A is equivalent to a proper subset of A.


7. A is enumerable (or, denumerable) if A ,...., J.


8. A is countable if A is either finite or denumerable.


9. A is uncountable if A is not countable.

Example Q is countable, whereas JR is uncountable.

Definitions 2.3 Let I and X be two nonempty sets. Consider a mapping

that associates with each element a of I a subset A"' of X. We shall call

I an index set, and the set of all sets Aa an indexed collection (or family)

of sets, denoted {Aa: a EI}, or briefly, {Aa}·

If the set I is the set JN = {1, 2, ... , N}, we use the notation {An}~

and this family of sets is called a finite sequence of sets. If I = J =

{1, 2, ... , n, ... }, we write {An};'° and this family is called an infinite se-

quence of sets. Often the term sequence is used to mean a finite or infinite

sequence, denoted {An}.

If the subsets An of X are singletons the sequence {An} is called a point

sequence or an ordinary sequence. It may be thought as defined by a single-

valued function mapping JN or J into X, and we refer to it as a sequence