applications of Urysohn’s Lemma to locally compact Hausdorff spaces

applications of Urysohn’s Lemma to locally compact Hausdorff spaces

Let

X

be a

locally compact Hausdorff space

(LCH space) and

X

*

its

one-point compactification

. We employ the following notation:

•

C

⁢

(

X

)

denotes the set of

continuous

complex functions on

X

;

•

C

b

⁢

(

X

)

denotes the set of continuous and

bounded

complex functions on

X

;

•

C

0

⁢

(

X

)

denotes the set of continuous complex functions on

X

which

vanish at infinity

;

•

C

c

⁢

(

X

)

denotes the set of continuous complex functions on

X

with compact support

Note that we have

C

c

⁢

(

X

)

⊆

C

0

⁢

(

X

)

⊆

C

b

⁢

(

X

)

⊆

C

⁢

(

X

)

, and that when we replace

X

with

X

*

(in general, when

X

is

compact

), these four classes of

functions

coincide.

Now, while Urysohn’s Lemma does not directly apply to

X

(since

X

need not in general be normal), it does apply to

X

*

, for being compact

Hausdorff

,

X

*

is

necessarily normal. One may therefore indirectly apply Urysohn’s Lemma to

X

by way of

X

*

to obtain various results asserting the

existence

of certain

continuous functions

on

X

with prescribed

properties

. The following results and their proofs illustrate this technique and are frequently useful in

analysis

.

Proposition 1

.

If

K

⊆

U

⊆

X

with

K

compact and

U

open, then there exists an

open subset

V

of

X

with compact closure such that

K

⊆

V

⊆

V

¯

⊆

U

.

Proof.

Since

K

is a compact subset of the Hausdorff space

X

*

, it is closed, and because

X

is open in

X

*

,

U

is as well. Therefore, by

normality

, there exists an open subset

V

of

X

*

such that

K

⊆

V

⊆

V

¯

⊆

U

(note that the

closure

of

V

in

X

*

coincides with that of

V

in

X

, since the former set is

contained

in

X

and the latter set is equal to the former intersected with

X

). As

V

¯

is closed in

X

*

, it is compact, and because

V

is open in

X

*

and

V

⊆

X

,

V

is open in

X

. Thus

V

possesses the desired properties.
∎

Corollary 1

.

For each

x

∈

X

and each open subset

U

of

X

containing

x

, there exists an open subset

V

of

X

with compact closure such that

x

∈

V

and

V

¯

⊆

U

.

Proof.

Take

K

=

{

x

}

in the preceding

proposition

.
∎

Theorem 1

.

(Urysohn’s Lemma for LCH Spaces)
If

K

⊆

U

⊆

X

with

K

compact and

U

open, then there exists

f

∈

C

c

⁢

(

X

)

such that

0

≤

f

≤

1

,

f

|

K

≡

1

, and

supp

⁡

f

⊆

U

.

Proof.

By the first Proposition, there exists an open subset

V

of

X

with compact closure such that

K

⊆

V

⊆

V

¯

⊆

U

; since

K

and

X

*

-

V

are

disjoint

closed subsets

of the

normal space

X

*

, Urysohn’s Lemma furnishes

g

∈

C

⁢

(

X

*

)

such that

0

≤

g

≤

1

,

g

|

K

≡

1

, and

g

|

X

*

-

V

≡

0

. Put

f

=

g

|

X

. Then

f

∈

C

⁢

(

X

)

,

0

≤

f

≤

1

, and

f

|

K

≡

1

. Moreover,

f

vanishes

outside

V

¯

because

g

does, so

{

x

∈

X

:

f

⁢

(

x

)

≠

0

}

⊆

V

¯

⊆

U

; since

V

¯

is compact, and consequently closed, the last

inclusion

gives

supp

⁡

f

⊆

V

¯

⊆

U

and

f

∈

C

c

⁢

(

X

)

.
∎

Theorem 2

.

(

Tietze Extension Theorem

for LCH Spaces)
If

K

⊆

X

is compact and

f

∈

C

⁢

(

K

)

is real, then there exists a real

g

∈

C

c

⁢

(

X

)

extending

f

.

Corollary 2

.

C

0

⁢

(

X

)

is the uniform closure of

C

c

⁢

(

X

)

in

C

b

⁢

(

X

)

.

Proof.

We first show that

C

0

⁢

(

X

)

is closed in

C

b

⁢

(

X

)

. To this end, assume that

(

f

n

)

n

=

1

∞

is a uniformly convergent

sequence

of functions in

C

0

⁢

(

X

)

with limit

f

and let

ϵ

>

0

be given. Select

N

∈

ℤ

+

such that

∥

f

-

f

N

∥

∞

<

ϵ

/

2

, and select a compact subset

K

of

X

such that

|

f

N

|

<

ϵ

/

2

for

x

∈

X

-

K

. We then have, for all such

x

,

|

f

⁢

(

x

)

|

=

|

f

⁢

(

x

)

-

f

N

⁢

(

x

)

+

f

N

⁢

(

x

)

|

≤

|

f

⁢

(

x

)

-

f

N

⁢

(

x

)

|

+

|

f

N

⁢

(

x

)

|

≤

∥

f

-

f

N

∥

∞

+

|

f

N

⁢

(

x

)

|

<

ϵ

⁢

.

Thus

f

vanishes at infinity; since the uniform limit of continuous functions is continuous, we obtain

f

∈

C

0

⁢

(

X

)

, whence

C

0

⁢

(

X

)

is closed. It remains to establish the

density

of

C

c

⁢

(

X

)

in

C

0

⁢

(

X

)

. Given

f

∈

C

0

⁢

(

X

)

and

ϵ

>

0

, select a compact subset

K

of

X

such that

|

f

⁢

(

x

)

|

<

ϵ

/

2

for

x

∈

X

-

K

. By

Theorem

1, there exists

g

∈

C

c

⁢

(

X

)

with range in

[

0

,

1

]

satisfying

g

|

K

≡

1

. The function

h

=

f

⁢

g

is continuous and supported inside

supp

⁡

g

, hence lies in

C

c

⁢

(

X

)

; moreover, if

x

∈

K

, then we have

|

f

⁢

(

x

)

-

h

⁢

(

x

)

|

=

|

f

⁢

(

x

)

-

f

⁢

(

x

)

|

=

0

, while if

x

∉

K

, then

|

f

⁢

(

x

)

-

h

⁢

(

x

)

|

=

|

f

⁢

(

x

)

-

f

⁢

(

x

)

⁢

g

⁢

(

x

)

|

=

|

f

⁢

(

x

)

|

⁢

|

1

-

g

⁢

(

x

)

|

≤

|

f

⁢

(

x

)

|

<

ϵ

2

⁢

.

It follows that

∥

f

-

h

∥

∞

<

ϵ

, hence that

f

∈

C

c

⁢

(

X

)

¯

, completing the proof.
∎

Title

applications of Urysohn’s Lemma to locally compact Hausdorff spaces

Canonical

name

ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

Date of creation

2013-03-22 18:33:31

Last modified on

2013-03-22 18:33:31

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

23

Author

azdbacks4234 (14155)

Entry type

Topic

Classification

msc 54D15

Related topic

UrysohnsLemma

Related topic

TietzeExtensionTheorem

Related topic

VanishAtInfinity

Related topic

SupportOfFunction

Related topic

LocallyCompact

Related topic

T2Space

Related topic

NormalTopologicalSpace

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