zero set of a topological space

zero set of a topological space

Let

X

be a

topological space

and

f

∈

C

⁢

(

X

)

, the

ring of continuous functions

on

X

. The

level set

of

f

at

r

∈

ℝ

is the set

f

-

1

⁢

(

r

)

:=

{

x

∈

X

∣

f

⁢

(

x

)

=

r

}

. The

zero set

of

f

is defined to be the level set of

f

at

0

. The zero set of

f

is denoted by

Z

⁢

(

f

)

. A subset

A

of

X

is called a

zero set

of

X

if

A

=

Z

⁢

(

f

)

for some

f

∈

C

⁢

(

X

)

.

Properties

. Let

X

be a topological space and, unless otherwise specified,

f

∈

C

⁢

(

X

)

.

1.

Any zero set of

X

is closed. The

converse

is not true. However, if

X

is a

metric space

, then any

closed set

A

is a zero set: simply define

f

:

X

→

ℝ

by

f

⁢

(

x

)

:=

d

⁢

(

x

,

A

)

where

d

is the metric on

X

.

2.

The level set of

f

at

r

is the zero set of

f

-

r

^

, where

r

^

is the

constant function

valued at

r

.

3.

Z

⁢

(

r

^

)

=

X

iff

r

=

0

. Otherwise,

Z

⁢

(

r

^

)

=

∅

. In fact,

Z

⁢

(

f

)

=

∅

iff

f

is a unit in the ring

C

⁢

(

X

)

.

4.

Since

f

⁢

(

a

)

=

0

iff

|

f

⁢

(

a

)

|

<

1

n

for all

n

∈

ℕ

, and each

{

x

∈

X

∣

|

f

⁢

(

x

)

|

<

1

n

}

is open in

X

, we see that

Z

⁢

(

f

)

=

⋂

n

=

1

∞

{

x

∈

X

∣

|

f

⁢

(

x

)

|

<

1

n

}

.

This shows every zero set is a

G

δ

(

http://planetmath.org/G_deltaSet

) set.

5.

For any

f

∈

C

⁢

(

X

)

,

Z

⁢

(

f

)

=

Z

⁢

(

f

n

)

=

Z

⁢

(

|

f

|

)

, where

n

is any positive integer.

6.

Z

⁢

(

f

⁢

g

)

=

Z

⁢

(

f

)

∪

Z

⁢

(

g

)

.

7.

Z

⁢

(

f

)

∩

Z

⁢

(

g

)

=

Z

⁢

(

f

2

+

g

2

)

=

Z

⁢

(

|

f

|

+

|

g

|

)

.

8.

{

x

∈

X

∣

0

≤

f

⁢

(

x

)

}

is a zero set, since it is equal to

Z

⁢

(

f

-

|

f

|

)

.

9.

If

C

⁢

(

X

)

is considered as an algebra over

ℝ

, then

Z

⁢

(

r

⁢

f

)

=

Z

⁢

(

f

)

iff

r

≠

0

.

The

complement

of a zero set is called a

cozero set

. In other words, a cozero set looks like

{

x

∈

X

∣

f

⁢

(

x

)

≠

0

}

for some

f

∈

C

⁢

(

X

)

. By the last property above, a cozero set also has the form

pos

⁡

(

f

)

:=

{

x

∈

X

∣

0

<

f

⁢

(

x

)

}

for some

f

∈

C

⁢

(

X

)

.

Let

A

be a subset of

C

⁢

(

X

)

. The

zero set

of

A

is defined as the set of all zero sets of elements of

A

:

Z

⁢

(

A

)

:=

{

Z

⁢

(

f

)

∣

f

∈

A

}

. When

A

=

C

⁢

(

X

)

, we also write

Z

⁢

(

X

)

:=

Z

⁢

(

C

⁢

(

X

)

)

and call it

the family of zero sets

of

X

. Evidently,

Z

⁢

(

X

)

is a subset of the family of all closed

G

δ

sets of

X

.

Remarks

.

•

By properties 6. and 7. above,

Z

⁢

(

X

)

is

closed under

set union and set

intersection

operations

. It can be shown that

Z

⁢

(

X

)

is also closed under

countable

intersections.

•

It is also possible to define a zero set of

X

to be the zero set of some

f

∈

C

*

⁢

(

X

)

, the subring of

C

⁢

(

X

)

consisting of the

bounded

continuous functions

into

ℝ

. However, this definition turns out to be

equivalent

to the one given for

C

⁢

(

X

)

, by the observation that

Z

⁢

(

f

)

=

Z

⁢

(

|

f

|

∧

1

^

)

.

References

1

L. Gillman, M. Jerison:

Rings of Continuous Functions

, Van Nostrand, (1960).

Title

zero set of a topological space

Canonical name

ZeroSetOfATopologicalSpace

Date of creation

2013-03-22 16:56:06

Last modified on

2013-03-22 16:56:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54C50

Classification

msc 54C40

Classification

msc 54C35

Defines

zero set

Defines

level set

Defines

cozero set

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