2Topological spaces

IB Metric and Topological Spaces

2.3 Closed sets

We will define closed sets similarly to what we did for metric spaces.

Definition (Closed sets). C ⊆ X is closed if X \C is an open subset of X.

Lemma.

(i) If C

α

is a closed subset of X for all α ∈ A, then

T

α∈A

C

α

is closed in X.

(ii) If C

1

, ··· , C

n

are closed in X, then so is

S

n

i=1

C

i

.

Proof.

(i)

Since

C

α

is closed in

X

,

X \ C

α

is open in

X

. So

S

α∈A

(

X \ C

α

) =

X \

T

α∈A

C

α

is open. So

T

α∈A

C

α

is closed.

(ii)

If

C

i

is closed in

X

, then

X \ C

i

is open. So

T

n

i=1

(

X \ C

i

) =

X \

S

n

i=1

C

i

is open. So

S

n

i=1

C

i

is closed.

This time we can take infinite intersections and finite unions, which is the

opposite of what we have for open sets.

Note that it is entirely possible to define the topology to be the collection of

all closed sets instead of open sets, but people seem to like open sets more.

Corollary. If X is Hausdorff and x ∈ X, then {x} is closed in X.

Proof.

For all

y ∈ X

, there exist open subsets

U

y

, V

y

with

y ∈ U

y

, x ∈ V

y

,

U

y

∩ V

y

= ∅.

Let

C

y

=

X \ U

y

. Then

C

y

is closed,

y 6∈ C

y

,

x ∈ C

y

. So

{x}

=

T

y6=x

C

y

is

closed since it is an intersection of closed subsets.