Definition 2.4.1A metric space is called separableif there exists a countabledense subset D. This means two things. First, D is countable, and second that if x isany point and r > 0, then B
(x,r)
∩ D≠∅. A metric space is called completely separableif there exists a countable collectionof nonempty open sets ℬ such that every open setis the union of some subset of ℬ. This collection of open 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 2.4.2A metric space is separable if and only if it is completelyseparable.
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
( δ)
y,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
. 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.
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Definition 2.4.3Let S be a nonempty set. Then a set of open sets C is calledan open cover of S if ∪C⊇S. (It covers up the set S. Think lilly pads covering thesurface of a pond.)
One of the important properties possessed by separable metric spaces is the Lindeloff
property.
Definition 2.4.4A metric space has the Lindeloff property if whenever C isan open cover of a set S, there exists acountable subset of C denoted here by ℬ suchthat ℬ is also an open cover of S.
Theorem 2.4.5Every 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 2.4.2. 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.
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Definition 2.4.6A Polish space is a complete separable metric space. Thesethings turn out to be very useful in probability theory and in other areas.