Download 1.2 Open Sets, Closed Sets, and Clopen Sets

Lecture note /topology / lecturer:Zahir Dobeas AL-nafie
1.2
Open Sets, Closed Sets, and Clopen Sets
Definition.
Let (X,
 ) be any topological space.
Then the members of

to be open sets.
Proposition.
If (X,
)
is any topological space, then
(i) X and Ø are open sets,
(ii) the union of any (finite or infinite) number of open sets is an open set and
(iii) the intersection of any finite number of open sets is an open set.
Proof.
Clearly (i) and (ii) are trivial consequences of Definition 1.2.1 and Definitions 1.1.1 (i)
and (ii). The condition (iii) follows from Definition 1.2.1 and Exercises 1.1 #4.
On reading Proposition 1.2.2, a question should have popped into your mind: while any
finite or infinite union of open sets is open, we state only that finite intersections of open sets
are open. Are infinite intersections of open sets always open? The next example shows that the
answer is "noÔ.
are said
2
Let N be the set of all positive integers and let

consist of Ø and each
Example.
subset S of N such that the complement of S in N, N \ S, is a finite set. It is easily verified that

satisfies Definitions 1.1.1 and so is a topology on N. (In the next section we shall discuss this
topology further. It is called the finite-closed topology.) For each natural number n, define the
set S n as follows:
Sn = {1}  {n + 1}  {n + 2}  {n + 3}  ··· = {1} 
Clearly each S n is an open set in the topology
,
[
{m}.

m =n+1
since its complement is a finite set. However,
\

Sn = {1}.
(1)
n=1
As the complement of {1} is neither N nor a finite set, {1} is not open. So (1) shows that the
intersection of the open sets S n is not open.
You might well ask: how did you find the example presented in Example 1.2.3? The answer
is unglamorous! It was by trial and error.
If we tried, for example, a discrete topology, we would find that each intersection of open
sets is indeed open. The same is true of the indiscrete topology. So what you need to do is some
intelligent guesswork.
Remember that to prove that the intersection of open sets is not necessarily open, you need
to find just one counterexample!
Definition.
closed set in (X,
Let (X,
)
)
be a topological space. A subset S of X is said to be a
if its complement in X, namely X \ S, is open in (X,
 ).
In Example , the closed sets are
•,X, {b,c,d,e,f}, {a,b,e,f}, {b,e,f}
 ) is a discrete space, then it is obvious that every subset of X
indiscrete space, (X,  ), the only closed sets are X and Ø.
If (X,
an
and {a}.
is a closed set. However in
. OPEN SETS
3
Proposition.
If (X,
)
is any topological space, then
(i) Ø and X are closed sets,
(ii) the intersection of any (finite or infinite) number of closed sets is a closed set and
(iii) the union of any finite number of closed sets is a closed set.
Proof.
(i) follows immediately from Proposition (i) and Definition , as the
complement of X is Ø and the complement of Ø is X.
To prove that (iii) is true, let S 1,S 2,...,S
n
be closed sets. We are required to prove that
S 1  S 2 ··· S n is a closed set. It sufces to show, by Definition 1.2.4, that X \ (S 1  S 2 ··· S n)
is an open set.
As S 1,S 2,...,S
n
are closed sets, their complements X \ S 1 , X \ S 2 , ..., X \ S n are open
sets. But
X \ (S 1  S 2  ···  S n) = (X \ S 1 )  (X \ S 2)  ···  (X \ S n).
As the right hand side of (1) is a finite intersection of open sets, it is an open set. So the
left hand side of (1) is an open set. Hence S 1  S 2  ···  S n is a closed set, as required. So (iii)
is true.
The proof of (ii) is similar to that of (iii).
(1)
4
(i) the set {a} is both open and closed;
(ii) the set {b,c} is neither open nor closed;
(iii) the set {c,d} is open but not closed;
(iv) the set {a,b,e,f} is closed but not open.
In a discrete space every set is both open and closed, while in an indiscrete space (X,
 ), all
subsets of X except X and Ø are neither open nor closed.
To remind you that sets can be both open and closed we introduce the following definition.
Definition.
A subset S of a topological space (X,
both open and closed in (X,
 ).
In every topological space (X,
)
both X and Ø are clopen 1.
In a discrete space all subsets of X are clopen.
)
is said to be clopen if it is