2Topological spaces

IB Metric and Topological Spaces

2.2 Sequences

To define the convergence of a sequence using open sets, we again need the

concept of open neighbourhoods.

Definition (Open neighbourhood). An open neighbourhood of

x ∈ X

is an open

set U ⊆ X with x ∈ U.

Now we can use this to define convergence of sequences.

Definition (Convergent sequence). A sequence

x

n

→ x

if for every open

neighbourhood U of x, ∃N such that x

n

∈ U for all n > N.

Example.

(i)

If

X

has the coarse topology, then any sequence

x

n

converges to every

x ∈ X, since there is only one open neighbourhood of x.

(ii)

If

X

has the cofinite topology, no two

x

n

s are the same, then

x

n

→ x

for every

x ∈ X

, since every open set can only have finitely many

x

n

not

inside it.

This looks weird. This is definitely not what we used to think of sequences.

At least, we would want to have unique limits.

Fortunately, there is a particular class of spaces where sequences are well-

behaved and have at most one limit.

Definition (Hausdorff space). A topological space

X

is Hausdorff if for all

x

1

, x

2

∈ X

with

x

1

6

=

x

2

, there exist open neighbourhoods

U

1

of

x

1

,

U

2

of

x

2

such that U

1

∩ U

2

= ∅.

Lemma. If

X

is Hausdorff,

x

n

is a sequence in

X

with

x

n

→ x

and

x

n

→ x

0

,

then x = x

0

, i.e. limits are unique.

Proof.

Suppose the contrary that

x 6

=

x

0

. Then by definition of Hausdorff, there

exist open neighbourhoods U, U

0

of x, x

0

respectively with U ∩ U

0

= ∅.

Since

x

n

→ x

and

U

is a neighbourhood of

x

, by definition, there is some

N

such that whenever

n > N

, we have

x

n

∈ U

. Similarly, since

x

n

→ x

0

, there is

some N

0

such that whenever n > N

0

, we have x

n

∈ U

0

.

This means that whenever

n > max

(

N, N

0

), we have

x

n

∈ U

and

x

n

∈ U

0

.

So x

n

∈ U ∩ U

0

. This contradicts the fact that U ∩ U

0

= ∅.

Hence we must have x = x

0

.

Example.

(i)

If

X

has more than 1 element, then the coarse topology on

X

is not

Hausdorff.

(ii)

If

X

has infinitely many elements, the cofinite topology on

X

is not

Hausdorff.

(iii) The discrete topology is always Hausdorff.

(iv)

If (

X, d

) is a metric space, the topology induced by

d

is Hausdorff: for

x

1

6

=

x

2

, let

r

=

d

(

x

1

, x

2

)

>

0. Then take

U

i

=

B

r/2

(

x

i

). Then

U

1

∩U

2

=

∅

.