properties of first countability

properties of first countability

Proposition 1

.

Let

X

be a

first countable

topological space

and

x

∈

X

. Then

x

∈

A

¯

iff there is a

sequence

(

x

i

)

in

A

that

converges

to

x

.

Proof.

One side is true for all topological spaces: if

(

x

i

)

is in

A

converging

x

, then for any

open set

U

of

x

, there is some

i

such that

x

i

∈

U

, whence

U

∩

A

≠

∅

. As a result,

x

∈

A

¯

.

Conversely

, suppose

x

∈

A

¯

. Let

{

B

i

∣

i

=

1

,

2

,

…

}

be a

neighborhood base

around

x

. We may as well assume each

B

i

open. Next, let

N

n

:=

B

1

∩

B

2

∩

⋯

∩

B

n

,

then we obtain a set of nested open sets containing

x

:

N

1

⊇

N

2

⊇

⋯

.

Since each

N

i

is open, its

intersection

with

A

is non-empty. So we may choose

x

i

∈

N

i

∩

A

. We want to show that

(

x

i

)

converges to

x

. First notice tat for any fixed

j

,

x

i

∈

N

j

for all

i

≥

j

. Pick any open set

U

containing

x

. Then

N

j

⊆

B

j

⊆

U

. Hence

x

i

∈

U

for all

i

≥

j

.
∎

From this, we can prove the following corollaries (assuming all spaces involved are first countable):

Corollary 1

.

C

is closed iff every sequence

(

x

i

)

in

C

that converges to

x

implies that

x

∈

C

.

Proof.

First, assume

(

x

i

)

is in a

closed set

C

converging to

x

. Then

x

∈

C

¯

by the

proposition

above. As

C

is closed, we have

x

∈

C

¯

=

C

.

Conversely, pick any

x

∈

C

¯

. By the proposition above, there is a sequence

(

x

i

)

in

C

converging to

x

. By

assumption

x

∈

C

. So

C

¯

⊆

C

, which means that

C

is closed.
∎

Corollary 2

.

U

is open iff every sequence

(

x

i

)

that converges to

x

∈

U

is eventually in

U

.

Proof.

First, suppose

U

is open and

(

x

i

)

converges to

x

∈

U

. If none of

x

i

is in

U

, then all of

x

i

is in its

complement

X

-

U

, which is closed. Then by the proposition,

x

must be in the

closure

of

X

-

U

, which is just

X

-

U

, contradicting the assumption that

x

∈

U

. Hence

x

i

∈

U

for some

i

.

Conversely, assume the right hand side statement. Suppose

x

∉

U

∘

=

X

-

X

-

U

¯

. Then

x

∈

X

-

U

¯

. By the proposition, there is a sequence

(

x

i

)

in

X

-

U

converging to

x

. If

x

∈

U

, then by assumption,

(

x

i

)

is eventually in

U

, which means

x

i

∈

U

for some

i

, contradicting the earlier statement that

(

x

i

)

is in

X

-

U

. Therefore,

x

∉

U

, which implies that

U

⊆

U

∘

, or

U

is open.
∎

Corollary 3

.

A

function

f

:

X

→

Y

is

continuous

iff it preserves converging sequences.

Proof.

Suppose first that

f

is continuous, and

(

x

i

)

in

X

converging to

x

. We want to show that

(

f

⁢

(

x

i

)

)

converges to

f

⁢

(

x

)

. Let

V

be an open set containing

f

⁢

(

x

)

. So

f

-

1

⁢

(

V

)

is open containing

x

, which implies that there is some

j

such that

x

i

∈

f

-

1

⁢

(

V

)

for all

i

≥

j

, or

f

⁢

(

x

i

)

∈

V

for all

i

≥

j

, which means that

(

f

⁢

(

x

i

)

)

→

f

⁢

(

x

)

.

Conversely, suppose

f

preserves converging sequences and

C

a closed set in

Y

. We want to show that

D

:=

f

-

1

⁢

(

C

)

is closed. Suppose

(

x

i

)

is a sequence in

D

converging to

x

. Then

(

f

⁢

(

x

i

)

)

converges to

f

⁢

(

x

)

. Since

(

f

⁢

(

x

i

)

)

is in

C

and

C

is closed,

f

⁢

(

x

)

∈

C

by the first corollary above. So

x

∈

f

-

1

⁢

(

C

)

=

D

too. Hence

D

is closed, again by the same corollary.
∎

Title

properties of first countability

Canonical name

PropertiesOfFirstCountability

Date of creation

2013-03-22 19:09:26

Last modified on

2013-03-22 19:09:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Result

Classification

msc 54D99

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