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
=
∅
.