Notations used in the
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
x e A
x g A
does not imply
implies and is implied by; if and only if
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