11.1.5 Separable Metric Spaces
Definition 11.1.28 A metric space is called separable if 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, then
∩ D≠∅. A metric space is called completely separable if there exists a countable collection of
nonempty open sets ℬ such that every open set is 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 11.1.29 A 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 pB be a point
of B. Then let P≡
is any ball, then it is the union of sets of
and so there is a
point of P
in it. Since ℬ
is countable, so is P
be the countable dense set and let ℬ≡
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
be a ball. Let
y ∈ B
Then there exists d ∈ B
Let ε ∈ ℚ
< ε <
y ∈ B
. Is B
? If so, then the desired result
follows because this would show that every
y ∈ B
is contained in one of these sets of
contained in B
is the union of sets of
. Let z ∈ B
. Therefore, every ball is the union of sets of
every open set is the union of balls, it follows that every open set is the union of sets of ℬ
Definition 11.1.30 Let S be a nonempty set. Then a set of open sets C is called an open cover
of 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 11.1.31 A metric space has the Lindeloff property if whenever C is an open cover of a
set S, there exists a countable subset of C denoted here by ℬ such that ℬ is also an open cover of S.
Theorem 11.1.32 Every separable metric space has 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.
denote those sets of
which are contained in some set of C
is a countable open cover of
Now for B ∈ℬ
, let UB
be a set of C
which contains B
denote these sets
it follows that
countable and is an open cover of
Definition 11.1.33 A Polish space is a complete separable metric space. These things turn out to
be very useful in probability theory and in other areas.