Supplement for Manifolds and Differential Geometry Jeffrey M. Lee

10 0. Background Material
Example 0.1. If R denotes the set of real numbers then, for a given positive
integer n, R n denotes the set of n-tuples of real numbers. This set has a lot
of structure as we shall see.
One of the most important notions in all of mathematics is the notion
of a “relation”. A relation from a set A to a set B is simply a subset of
A × B. A relation from A to A is just called a relation on A. If R is a
relation from A to B then we often write a R b instead of (a, b) ∈ R. We
single out two important types of relations:
Definition 0.2. An equivalence relation on a set X is a relation on X,
usual denoted by the symbol ∼, such that (i) x ∼ x for all x ∈ X, (ii) x ∼ y
if and only if y ∼ x, (iii) if x ∼ y and y ∼ z then x ∼ z. For each a ∈ X the
set of all x such that x ∼ a is called the equivalence class of a (often denoted
by [a]). The set of all equivalence classes form a partition of X. The set of
equivalence classes if often denoted by X/∼.
Conversely, it is easy to see that if {A i } i∈I is a partition of X then we
my defined a corresponding equivalence relation by declaring x ∼ y if and
only if x and y belong to the same A i .
For example, ordinary equality is an equivalence relation on the set of
natural numbers. Let Z denote the set of integers. Then equality modulo
a fixed integer p defines and equivalence relation on Z where n ∼ m iff 1
n − m = kp for some k ∈ Z. In this case the set of equivalence classes is
denoted Z p or Z/pZ.
Definition 0.3. A partial ordering on a set X (assumed nonempty) is
a relation denoted by, say ≼, that satisfies (i) x ≼ x for all x ∈ X, (ii) if
x ≼ y and y ≼ x then x = y , and (iii) if x ≼ y and y ≼ z then x ≼ z. We
say that X is partially ordered by ≼. If a partial ordering also has the
property that for every x, y ∈ X we have either x ≼ y or y ≼ x then we call
the relation a total ordering or (linear ordering). In this case, we say that
X is totally ordered by ≼.
Example 0.4. The set of real numbers is totally ordered by the familiar
notion of less than; ≤.
Example 0.5. The power set P(X) is partially ordered by set inclusion ⊂
(also denoted ⊆).
If X is partially ordered by ≼ then an element x is called a maximal
element if x ≼ y implies x = y. A minimal element is defined similarly.
1 “iff” means ”if and only if”.