Basic Analysis – Gently Done Topological Vector Spaces

4. Separation
The Hausdorff property of a topological space is the ability to separate distinct
points. We shall extend this idea and consider the separation of disjoint closed
sets.
Definition 4.1 A topological space (X,T) is said to be normal if for any pair of
disjoint closed sets A and B there exist disjoint open sets U and V such that
A ⊆ U and B ⊆ V.
In other words, a topological space is normal if and only if disjoint closed
sets can be separated by disjoint open sets. If we extend the use of the word
neighbourhood by saying that N is a neighbourhood of a set C if there is some
open set G such that C ⊆ G ⊆ N, then (X,T) is normal if and only if every pair
of disjoint closed sets have disjoint neighbourhoods.
If (X,T) is normal and if every one-point set is closed, then (X,T) is Hausdorff.
For this reason, the condition that one-point sets be closed is often taken as part
of the definition of a normal topological space.
Example 4.2 Let X = {0,1,2} with T = � ∅,X,{0},{1,2} � . Then (X,T) is
normal—every closed set is also open. However, (X,T) is not Hausdorff. In this
case, the one-point sets {1} and {2} are not closed.
Let B(x;r) denote the ‘open’ ball B(x;r) = {x ′ ∈ X : d(x,x ′ ) < r} in the
metric space (X,d).
Proposition 4.3 Every metrizable topological space is normal.
Proof Suppose that A and B are non-empty disjoint closed sets in the metrizable
space (X,T). Then X\A and X\B are open sets with B ⊆ X\A and A ⊆ X\B.
Hence, for each a ∈ A, there is some ε a > 0 such that B(a;ε a ) ⊆ X \ B, and
similarly, for each b ∈ B, there is some ε b > 0 such that B(b;ε b ) ⊆ X\A. It follows
that d(a,b) ≥ max{ε a ,ε b }, for any a ∈ A, b ∈ B. For a ∈ A, set U a = B(a; 1
2 ε a ),
and for b ∈ B, set V b = B(b; 1
2 ε b
). Put U = �
a∈A U a
and V are both open and A ⊆ U and B ⊆ V.
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and V = �
b∈B V b
. Then U