Notations used in the

text

The three digit system is used to number the definitions, theorems, propo-

sitions, lemmas, exercises, notes and remarks. For example, Theorem 3.2.4

is the 4th numbered statement in section 2 of chapter 3.

The symbol • is used to indicate the end of a proof. The symbol

A := B or B =: A means that this equality is the definition of A by B. The

symbol • before an exercise means that this exercise will be needed in the

later discussions. Sections, theorems, propositions, etc., which are marked

* can be omitted on first reading.

The phrase "the following are equivalent:" means each of the listed state-

ment implies the other. For example in Theorem 1.1.4, it means that each

of the statements (i), (ii) and (iii) implies the other.

The notations and symbols used from logic and elementary analysis are

as follows:

7 ^

3

V

x e A

x g A

AcB

ACB

V{X)

implies; gives

does not imply

implies and is implied by; if and only if

there exists

for all; for every

x belongs to A

x does not belong to A

A is a proper subset of B

A is a subset of B

set of all subsets of X

xxi