alternative characterizations of Noetherian topological spaces, proof of

alternative characterizations of Noetherian topological spaces, proof of

We prove the equivalence of the following five conditions for a

topological space

X

:

•

(DCC)

X

satisfies

the

descending chain condition

(

http://planetmath.org/DescendingChainCondition

) for

closed subsets

.

•

(ACC)

X

satisfies the

ascending chain condition

(

http://planetmath.org/AscendingChainCondition

) for

open subsets

.

•

(Min) Every nonempty family of closed subsets has a

minimal element

.

•

(Max) Every nonempty family of open subsets has a

maximal element

.

•

(HC) Every subset of

X

is

compact

.

Proof

.

Let

f

:

P

⁢

(

X

)

→

P

⁢

(

X

)

be the

complement

map,
i.e.,

f

⁢

(

A

)

=

X

∖

A

for any subset

A

of

X

.
Then

f

induces an

order-reversing

bijective

map
between the open subsets of

X

and the closed subsets of

X

.
Sending arbitrary chains (

http://planetmath.org/TotalOrder

)/sets
of open/closed subsets of

X

through

f

then immediately yields the equivalence of conditions of the

theorem

, and also the equivalence of (Min) and (Max).

(Min)

⇒

(DCC) is obvious, since the elements of
an

infinite

strictly descending chain of closed subsets of

X

would form a set of closed subsets of

X

without minimal element.
Likewise, given a non-empty set

S

consisting of closed subsets of

X

without minimal element,
one can

construct

an infinite strictly descending chain in

S

simply by starting with any

A

0

∈

S

and choosing for

A

n

+

1

any

proper subset

of

A

n

satisfying

A

n

+

1

∈

S

.
Hence, we have proven conditions (ACC), (DCC), (Min) and (Max) of the theorem
to be

equivalent

,
and will be done if we can prove equivalence of statement (HC) to the others.

To this end, first assume statement (HC),
and let

(

U

i

)

i

∈

ℕ

be an ascending

sequence

of open subsets of

X

.
Then obviously, the

U

i

form an

open cover

of

U

=

⋃

i

∈

ℕ

U

i

,
which by

assumption

is bound to have a finite subcover.
Hence, there exists

n

∈

ℕ

such that

⋃

i

=

0

n

U

i

=

⋃

i

∈

ℕ

U

i

,
so our ascending sequence is in fact

stationary

.

Conversely

, assume statement (Max) of the theorem,
let

A

⊆

X

be any subset of

X

and let

(

U

i

)

i

∈

I

be a family of open sets in

X

such that the

U

i

∩

A

form an open cover of

A

with respect to the

subspace topology

.
Then, by the assumption,
the set of finite unions of the

U

i

has at least one maximal element,
say

U

,
and with any

i

∈

I

we obtain

U

i

∪

U

=

U

because of maximality of

U

.
Hence, we have

U

i

⊆

U

for all

i

∈

I

,
so in fact

⋃

i

∈

I

U

i

=

U

.
But,

U

was a union of a finite number of

U

i

by construction;
hence, a finite subcovering of

U

and thereby of

A

has been found. ∎

Title

alternative characterizations of Noetherian topological spaces

, proof of

Canonical name

AlternativeCharacterizationsOfNoetherianTopologicalSpacesProofOf

Date of creation

2013-03-22 15:25:39

Last modified on

2013-03-22 15:25:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Proof

Classification

msc 14A10

Generated on Fri Feb 9 17:56:20 2018 by

LaTeXML