Definition 11.1.28A metric space is called separableif there exists a countable dense subset D.This means two things. First, D is countable, and second that if x is any point and r > 0, thenB
(x,r)
∩ D≠∅. A metric space is called completely separableif there exists a countable collectionofnonempty open sets ℬ such that every open set is the union of some subset of ℬ. This collection ofopen sets is called a countable basis.
For those who like to fuss about empty sets, the empty set is open and it is indeed the union of a subset of
ℬ namely the empty subset.
Theorem 11.1.29A metric space is separable if and only if it is completely separable.
Proof: ⇐= Let ℬ be the special countable collection of open sets and for each B ∈ℬ, let p_{B} be a point
of B. Then let P≡
{pB : B ∈ ℬ}
. If B
(x,r)
is any ball, then it is the union of sets of ℬ and so there is a
point of P in it. Since ℬ is countable, so is P.
=⇒
Let D be the countable dense set and let ℬ≡
{B (d,r) : d ∈ D,r ∈ ℚ ∩ [0,∞ )}
. Then ℬ is
countable because the Cartesian product of countable sets is countable. It suffices to show that every ball is
the union of these sets. Let B
(x,R)
be a ball. Let y ∈ B
(y,δ)
⊆ B
(x,R)
. Then there exists d ∈ B
(y, δ)
10
.
Let ε ∈ ℚ and
δ-
10
< ε <
δ
5
. Then y ∈ B
(d,ε)
∈ℬ. Is B
(d,ε)
⊆ B
(x,R )
? If so, then the desired result
follows because this would show that every y ∈ B
(x,R )
is contained in one of these sets of ℬ which is
contained in B
(x,R)
showing that B
(x,R )
is the union of sets of ℬ. Let z ∈ B
(d,ε)
⊆ B
(d, δ)
5
.
Then
δ δ δ
d(y,z) ≤ d (y,d)+ d (d,z) < 10 + ε < 10 + 5 < δ
Hence B
(d,ε)
⊆ B
(y,δ)
⊆ B
(x,r)
. Therefore, every ball is the union of sets of ℬ and, since
every open set is the union of balls, it follows that every open set is the union of sets of ℬ.
■
Definition 11.1.30Let S be a nonempty set. Then a set of open sets C is called an open coverof S if ∪C⊇S. (It covers up the set S. Think lilly pads covering the surface of a pond.)
One of the important properties possessed by separable metric spaces is the Lindeloff property.
Definition 11.1.31A metric space has the Lindeloff property if whenever C is an open cover of aset S, there exists acountable subset of C denoted here by ℬ such that ℬ is also an open cover of S.
Theorem 11.1.32Every separable metric spacehas the Lindeloff property.
Proof: Let C be an open cover of a set S. Let ℬ be a countable basis. Such exists by Theorem 11.1.29.
Let
ˆℬ
denote those sets of ℬ which are contained in some set of C. Thus
ˆℬ
is a countable open cover of S.
Now for B ∈ℬ, let U_{B} be a set of C which contains B. Letting
^C
denote these sets U_{B} it follows that
^C
is
countable and is an open cover of S. ■
Definition 11.1.33A Polish space is a complete separable metric space. These things turn out tobe very useful in probability theory and in other areas.