72 Topology

CHAPTER 5

Separation Axioms

The character, in particular richness of a topological space, depends much upon the abundance of opensets in its topology. In fact, the more is the number of open sets, the stronger is the topology and thenhigher is the chance of any function defined on it being continuous. The study of separation axiomsinitiated by Alexandroff and Hopf provides us a tool of analyzing the strength of various topologicalspaces.

5.1 THE AXIOMS

Let (X, T ) be a topological space.The separation axioms as per Alexandroff and Hopf are as follows:

T1 -axiom: For every pair of distinct points x, y of X, there exist open sets G and H of X such that

x G y G∈ /∈, and y H x H∈ /∈, .

T2 -axiom: For every pair of distinct points x, y of X, there exist open sets G and H of X such that x G∈and y H G H∈ ∩ =, φ .

T3 -axiom: For every point x of X and every closed set F of X not containing x, there exist open sets G

and H of X such that x G∈ and F H G H⊂ ∩ =, φ .

T4 -axiom: For every pair of disjoint closed sets F and K of X, there exist open sets G and H of X such

that F G⊂ and K H G H⊂ ∩ =, φ .

Definition: A topological space (X, T) is called a T1 -space if it satisfies the T1 -axiom.

First observe the following:

Theorem 5.1.1: A topological space (X, T) is a T1 -space iff every singleton subset of X is closed.

x x

x

y y F

T1-axiom T2-axiom T3-axiom

Separation Axioms 73

Proof: Let (X, T ) be a T1 -space and let p be an arbitrary point of X. We shall show that { }p c is open.

To this end let x p c∈{ } . Now x and p being distinct points there exist open sets Gx and Hx such that

x Gx∈ and p Gx/∈ . Let G G x pxc= ∪ ∈{ ; { } } . Clearly G is open and G p c= { } . So {p} is closed.

Conversely, let x and y be two distinct points of X. Then {x} and {y} are two closed sets. So { }x c

and { }y c are two open sets satisfying the conditions that y x c/∈{ } and x y c/∈{ } . Hence (X, T) satisfies

the T1 -axiom, i.e., X is a T1-space.

Corollary: Every finite subset of a T1 -space is closed.

Definition: A topological space (X, T ) is a called a T2 -space or a Hausdorff space if it satisfies the T2 -axiom of separation.

Note the following then:

(a) Every Hausdorff space is a T1-space.(b) Every metric space is a Hausdorff space.(c) Every subspace of Hausdorff space is Hausdorff.(d) The cartesian product of Hausdorff spaces is Hausdorff.

(e) The set {( ); }x x x X, ∈ is closed in X X× if X is Hausdorff.

(f) Every convergent sequence in a Hausdorff space converges to a unique limit.

An interesting at the same time important result about Hausdorff spaces is the following:

Theorem 5.1.2: If ( )X TX, be a topological space and ( )Y TY, be a Hausdorff space and f g X Y, : →are continuous functions, then

(i) The set { ; ( ) ( )}x X f x g x, ∈ = is closed in X

(ii ) f(x) = g(x) for all x in D, D is dense in X implies f(x) = g(x) for all x in X.

(iii ) The set graphf x f x x X= ∈{( ( )); }, is closed in X Y× .

Proof: (i) Since { ; ( ) ( )} ( )x X f x g x∈ = = −ψ 1 ∆ where ψ : X Y Y→ × defined by

ψ ( ) ( ( ) ( ))x f x g x= , is continuous and ∆ = ∈{( ) ; }y y y Y, , the result follows:

(ii ) Since by (i) D is closed and by the given condition D is dense, X D D= = . Hence the

result.

(iii ) Note graph f = −ϕ 1( )∆ where ϕ: X Y Y Y× → × defined by ϕ ( ) ( ( ) )x y f x y, ,= iscontinuous. Hence the result follows:

Definition: A topological space (X, T) is called a regular space if it satisfies the T3 -space.

X x

f x( )

74 Topology

First note that a regular space need not that be a T1 -space and hence need not be a Hausdorff space.

This is clear as the space (X, T) with X = {a, b, c}, T a b c X= { { } { } }φ, , , , is regular but not T1 -space(note the singleton set is {b} is closed}.

Definition: A topological space (X, T) is called a T3 -space if it is regular and T1 .

Observe that every T3 -space is T2 -space b but not conversely. Since a T3 -space is also T1 -space,every singleton set is closed and therefore for any two distinct points x and y of X, {y} being closed,

there exist two open sets G and H separating x and {y}, i.e., x G∈ , { }y H G H⊂ ∩, , but this is exactlythe requirement of a Hausdorff space.

A Hausdorff space which is not regular is the space (R, U) where T is the topology generated by allopen intervals and Q. Evidently T U⊃ and hence R is Hausdorff but it is not regular since the closed

set Qc and the point 1 /∈Qc can not be separated by disjoint open sets of T.

Other important results about regular spaces are

(i) Every subspace of a regular space is regular,(ii ) The cartesian product of regular spaces is regular.

Let C(X, R) denote the set of all bounded continuous real functions defined on X ∈. A family A of

functions from C(X, R) is said to separate points of X if for every pair of distinct points x, y of X, there

exist f A∈ such that f x f y( ) ( )≠ .

It readily follows that (X, T) is Hausdorff if C(X, R) separates points of X. This is clear, as for x < y,

there exist f C X∈ ( ), R and r ∈R such that f(x) < r < f(y) or f(y) < r < f(x). Evidently in the first case{ ; ( ) }x X f x r∈ < and { ; ( ) }x X f x r∈ > are disjoint open sets of X containing x and y respectively.Similarly for the second case we can find disjoint open sets separating x and y.

Definition: A topological space (X, T) is called a completely regular space if it satisfies the property that

for every x in X and closed set F of X, not containing x, there exist f C X∈ ( [ ]), ,0 1 such that f(x) = 0

and f(F) = 1.

A completely regular space is sometimes referred to as a T312 space or Tychonoff space.

From the definition it follows that

(i) Every completely regular space is a regular space. This is evident as for a point x of X and aclosed set not containing x, there exists f C X∈ ( [ ]), ,0 1 such that f(x) = 0 and f(F) = 1.

Now consider G x X f x= ∈ <{ ; ( ) }1 2/ and H x X f x= ∈ >{ ; ( ) }1 2/ . Clearly G and Hare disjoint open sets containing x and F respectively.

(ii ) For every x and a closed set F of X, there exists a function g C X∈ ( [ ]), ,0 1 such that g(x) =

1 and f(F) = 0.(iii ) C(X, [0, 1] separates points of X if X is completely regular.

(iv) Every Tychonoff space is a T3 -space.

Separation Axioms 75

Definition: A topological space is called normal if it satisfies the T4 -axiom and is a T1 -space, i.e., for

every pair of disjoint closed sets F and G, there exist disjoint open sets G and H of X such that

F G K H⊂ ⊂, .

A normal T1 -space is called a T4 -space.

The following results directly from the definition:

(i) Every metric space is normal

(ii ) A normal sapce need not be a T1 -space. Consider the space (X, T) where X = {a, b, c} and

T a b a b X= { { } { } { } }φ, , , , , . Clearly X is normal but it is not a T1 -space as {a} is not closed.

(iii ) Every T4 -space is a T3 -space and hence Hausdorff also.

5.2 URYSHON’S LEMMA AND TIETZE’S EXTENSION THEOREM

The fact that a topological space is rich in open sets ensures that it is rich in continuous functions. Thisis assured by the following results:

Urysohn’s Lemma: If (X, T) is a normal space and F and K are two closed subsets of X, then there

exists f C X∈ ( [ ]), ,0 1 such that f(F) = 0 and f(K) = 1.

The proof of this theorem is bit lengthy and is therefore left here but can be seen in [9].This lemma however leads to the following very useful result.

Tietze’s Extension Theorem: If X be a normal space and F be a closed subset of X equipped with therelative topology, then every continuous function from F to [a, b] admits of a continuous extension to X.

For a proof the reader is referred to [9].Note the condition of closedness of F is an unavoidable condition for the conclusion as can be seen

from the following example:Let X = [0, 1], F = (0, 1] and f(x) = (1/x). Clearly X is normal, F is not closed, f is continuous on F,

but f does not admit of a continuous extension to [0, 1].We conclude this section with the statement of a remarkable result on metrization.

Urysohn’s Metrization Theorem: Every second countable normal T1-space is homeomorphic a subset

of the Hilbert Cube I ∝ .

Recall the Hilbert cube I x x x x nn∝ ∝= ∈ <{( ) ; }1 2 3 1, , , | |K R / endowed with the relativized usual

topology.