Theory of Statistics - George Mason University

0.0 Some Basic Mathematical Concepts 619
in terms of the metric. The topology provides the definition of a closed set, as
above; that is, a set A ⊆ Ω is said to be closed iff Ω ∩ A c ∈ T , where T is the
collection of open sets defined in terms of the metric. As with the definitions
above for general topological spaces, some sets are both open and closed, and
such sets are said to be clopen.
We note that (IR, ρ) is a Hausdorff space because, given x, y ∈ IR and
x = y we have ρ(x, y) > 0 and so N(x, ρ(x, y)/2) and N(y, ρ(x, y)/2) are
disjoint open sets.
We also note that IR is connected, as is any interval in IR. (Connectedness
is a topological property that is defined on page 617.)
We will defer further discussion of openness and related concepts to
page 639 in Section 0.0.5 where we discuss the real number system.
0.0.2.9 Relations and Functions
A relation is a set of doubletons, or pairs of elements; that is, a relation is a
subset of a cartesian product of two sets. We use “relation” and “mapping”
synonymously.
A function is a relation in which no two different pairs have the same first
element.
To say that f is a function from Ω to Λ, written
f : Ω ↦→ Λ,
means that for every ω ∈ Ω there is a pair in f whose first member is ω. We
use the notation f(ω) to represent the second member of the pair in f whose
first member is ω, and we call ω the argument of the function. We call Ω the
domain of the function and we call {λ|λ = f(ω) for some ω ∈ Ω} the range
of the function.
Variations include functions that are onto, meaning that for every λ ∈ Λ
there is a pair in f whose second member is λ; and functions that are one-toone,
often written as 1 : 1, meaning that no two pairs have the same second
member. A function that is one-to-one and onto is called a bijection.
A function f that is one-to-one has an inverse, written f −1 , that is a
function from Λ to Ω, such that if f(ω0) = λ0, then f −1 (λ0) = ω0.
If (a, b) ∈ f, we may write a = f −1 (b), although sometimes this notation is
restricted to the cases in which f is one-to-one. If f is not one-to-one and if the
members of the pairs in f are reversed, the resulting relation is not a function.
We say f −1 does not exist; yet for convenience we may write a = f −1 (b), with
the meaning above.
If A ⊆ Ω, the image of A, denoted by f[A], or just by f(A), is the set of all
λ ∈ Λ for which λ = f(ω) for some ω ∈ Ω. (The notation f[A] is preferable,
but we will often just use f(A).) Similarly, if C is a collection of sets (see
below), the notation f[C] denotes the collection of sets {f[C] : C ∈ C}.
For the function f that maps from Ω to Λ, Ω is called the domain of the
function and and f[Ω] is called the range of the function.
Theory of Statistics c○2000–2013 James E. Gentle