relationship among different kinds of compactness

relationship among different kinds of compactness

The goal of this article is to prove

Theorem 1

.

If

X

is

second countable

and

T

1

, or if

X

is a

metric space

, then the following are

equivalent

:

1.

X

is

compact

;

2.

X

is

limit point compact

;

3.

X

is

sequentially compact

.

We prove this using several subsidiary

theorems

, which prove the various

implications

in

stronger

settings.

Theorem 2

.

A compact

topological space

T

is limit point compact. (Here we make no

assumptions

about the

topology

on

T

).

Proof.

Choose an subset

A

⊂

T

, and suppose

A

has no

limit points

. Then

A

contains its (

vacuous

set of) limit points and is therefore closed. But

closed subsets

of

compact spaces

are compact, so

A

is compact. Since

A

has no limit points, we may choose a

neighborhood

U

a

of each

a

∈

A

such that

U

a

intersects

A

only in

a

. But this cover clearly has a finite

subcover

only if

A

is finite. So any set without limit points is finite, and thus any

infinite set

has a limit point. This concludes the proof.

Theorem 3

.

If

T

is

first countable

,

T

1

, and limit point compact, then

T

is sequentially compact.

Proof.

Let

x

i

be any

sequence

of points in

T

, and assume that

x

i

takes infinitely many values (otherwise it obviously has a

convergent

subsequence

). Choose a limit point

x

for the sequence; we may assume wlog that

x

i

is equal to

x

for only finitely many

i

(otherwise again the result holds trivially). So by ignoring a finite number of leading terms of the sequence, we may assume that

x

i

≠

x

for every

i

. Since

T

is first countable, choose a countable basis

B

i

at

x

; by replacing

B

n

with

B

1

∩

…

∩

B

n

, we may assume that

B

i

+

1

⊂

B

i

for all

i

.

Now, choose

n

1

such that

p

n

1

∈

B

1

. Inductively, assume we have chosen

n

1

,

…

,

n

k

with

p

n

k

∈

B

k

. Since

T

is

T

1

, we may choose a neighborhood

U

of

q

that is

disjoint

from

p

n

1

,

…

,

p

n

k

; choose

p

n

k

+

1

to be any point in

U

∩

B

k

+

1

. Then inductively the

p

n

i

form a subsequence with

p

n

i

∈

B

i

, and clearly the

p

n

i

converge

to

q

. This concludes the proof.

Note that every metric space and every second countable

T

1

space is also first countable and

T

1

.

Proposition 1

.

Any sequentially compact metric space

M

is second countable.

Proof.

It clearly suffices to show that

M

has a

countable

dense subset

.

Claim first that for

ϵ

>

0

, the set of

ϵ

-balls in

M

has a finite subcover. Suppose this is false for some particular

ϵ

. Let

p

1

∈

M

be any point, and construct inductively points

p

k

with

p

k

∉

B

ϵ

⁢

(

p

1

)

∪

…

⁢

B

ϵ

⁢

(

p

k

-

1

)

. Since

M

is sequentially compact, we may replace the

p

i

by a convergent subsequence, which we also call

p

i

, with

p

i

→

p

∈

M

. But

convergent sequences

are Cauchy, so for

n

large enough, we have

d

⁢

(

p

n

,

p

m

)

<

ϵ

, which contradicts the construction of the

p

i

. This proves the claim.

Then for each

positive integer

n

, let

p

n

,

n

1

,

…

,

p

n

,

n

k

∈

M

be a

finite set

of points such that the

1

n

-balls around those points cover

M

. This set of points is countable, and is obviously

dense in

M

. This concludes the proof.

Theorem 4

.

If

T

is second countable or is a metric space, and sequentially compact, then

T

is compact.

Proof.

Assume first that

T

is second countable. Choose any open cover of

T

; it has a countable subcover

U

i

. We use an

argument

very similar to that used in the above

proposition

. Suppose no finite subset of the

U

i

covers

T

, and choose

p

k

∈

T

\

(

U

1

∪

…

∪

U

k

)

. Since

T

is sequentially compact, the

p

k

have a convergent subsequence

p

n

k

converging to

p

∈

T

. But

p

∈

U

n

for some

n

; since the

p

n

k

converge to

p

, all

p

n

k

∈

U

n

for

k

large enough. But this is a

contradiction

to the construction of the

p

k

, so that a finite subset of the

U

i

cover

T

and

T

is compact.

Since any sequentially compact metric space is second countable by the above proposition, we are done.

The main theorem follows trivially from the above. Note that we have in fact proven the following set of implications:

•

Compact

⇒

limit point compact for general topological spaces;

•

Limit point compact

⇒

sequentially compact for first countable

T

1

spaces;

•

Sequentially compact

⇒

compact for second countable or

metrizable spaces

.

Title

relationship among different kinds of compactness

Canonical name

RelationshipAmongDifferentKindsOfCompactness

Date of creation

2013-03-22 18:00:57

Last modified on

2013-03-22 18:00:57

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 54D30

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