6.1.2 Topological Spaces and Their Building Blocks
On the basis of the aforementioned discussion, we define a topology and topological
space next. Let T be a universal set. Let τ be a collection of subsets of T. If the
collection τ satisfies the following axioms, τ is called a topology on T. The relevant
axioms are as follows:
(O1) T 2 τ and ∅ 2 τ, where ∅ is an empty set.
(O2) If O 1 , O 2 , Á Á Á, O n 2 τ, O 1 \ O 2 Á Á Á \ O n 2 τ.
(O3) If O λ (λ 2 Λ) 2 τ, [ λ 2 Λ O λ 2 τ.
In the above axioms, T is called an underlying set. The coupled object of T and τ
is said to be a topological space and denoted by (T, τ). The members (i.e., subsets) of
τ are defined as open sets. (If we do not assume the topological space, the underlying
set is equivalent to the universal set. Henceforth, however, we do not have to be too
strict with the difference in this kind of terminology.)
Let (T, τ) be a topological space with a subset O
0
⊂ T. Let τ
0 be defined such that
τ
0
O
0
\ O; O 2 τ
f
g :
Then, (O
0 , τ
0 ) can be regarded as a topological space. In this case τ
0 is called a
relative topology for O
0 [2, 3] and O
0 is referred to as a subspace of T. The topological
space (O
0 , τ
0 ) shares the same topological feature as that of (T, τ). Notice that O
0 does
not necessarily need to be an open set of T. But, if O
0 is an open set of T, then any
open set belonging to O
0 is an open set of T as well. This is evident from the
definition of the relative topology and Axiom (O2).
The abovementioned Axioms (O1)–(O3) sound somewhat pretentious and the
definition of the open sets seems to be “descent from heaven.” Nonetheless, these
axioms and terminology turn out useful soon. Once a topological space (T, τ) is
given, subsets of various types arise and a variety of relationships among them ensue
as well. Note that the above axioms mention nothing about a set that is different from
an open set. Let S be a subset (maybe an open set or maybe not) of T and let us think
of the properties of S.
Definition 6.1 Let (T, τ) be a topological space and S be an open set of τ; i.e., S 2 τ.
Then, a subset defined as a complement of S in (T, τ); i.e., S
c ¼ T À S is called a
closed set in (T, τ).
Replacing S with S
c in Definition 6.1, we have (S
c
)
c
¼ S ¼ T À S
c . The above
definition may be rephrased as follows: Let (T, τ) be a topological space and let A be
a subset of T. Then, if A
c
¼ T À A 2 τ, then A is a closed set in (T, τ).
In parallel with the axioms (O1) to (O3), we have the following axioms related to
the closed sets of (T, τ): Let e τ be a collection of all the closed sets of (T, τ).
(C1) T 2 e τ and ∅ 2 e τ, where ∅ is an empty set.
(C2) If C 1 , C 2 , Á Á Á, C n 2 e τ, C 1 [ C 2 Á Á Á [ C n 2 e τ.
(C3) If C λ λ 2 Λ
ð
Þ2e τ, \ λ2Λ C λ 2 e τ.
6.1 Set and Topology
185
On the basis of the aforementioned discussion, we define a topology and topological
space next. Let T be a universal set. Let τ be a collection of subsets of T. If the
collection τ satisfies the following axioms, τ is called a topology on T. The relevant
axioms are as follows:
(O1) T 2 τ and ∅ 2 τ, where ∅ is an empty set.
(O2) If O 1 , O 2 , Á Á Á, O n 2 τ, O 1 \ O 2 Á Á Á \ O n 2 τ.
(O3) If O λ (λ 2 Λ) 2 τ, [ λ 2 Λ O λ 2 τ.
In the above axioms, T is called an underlying set. The coupled object of T and τ
is said to be a topological space and denoted by (T, τ). The members (i.e., subsets) of
τ are defined as open sets. (If we do not assume the topological space, the underlying
set is equivalent to the universal set. Henceforth, however, we do not have to be too
strict with the difference in this kind of terminology.)
Let (T, τ) be a topological space with a subset O
0
⊂ T. Let τ
0 be defined such that
τ
0
O
0
\ O; O 2 τ
f
g :
Then, (O
0 , τ
0 ) can be regarded as a topological space. In this case τ
0 is called a
relative topology for O
0 [2, 3] and O
0 is referred to as a subspace of T. The topological
space (O
0 , τ
0 ) shares the same topological feature as that of (T, τ). Notice that O
0 does
not necessarily need to be an open set of T. But, if O
0 is an open set of T, then any
open set belonging to O
0 is an open set of T as well. This is evident from the
definition of the relative topology and Axiom (O2).
The abovementioned Axioms (O1)–(O3) sound somewhat pretentious and the
definition of the open sets seems to be “descent from heaven.” Nonetheless, these
axioms and terminology turn out useful soon. Once a topological space (T, τ) is
given, subsets of various types arise and a variety of relationships among them ensue
as well. Note that the above axioms mention nothing about a set that is different from
an open set. Let S be a subset (maybe an open set or maybe not) of T and let us think
of the properties of S.
Definition 6.1 Let (T, τ) be a topological space and S be an open set of τ; i.e., S 2 τ.
Then, a subset defined as a complement of S in (T, τ); i.e., S
c ¼ T À S is called a
closed set in (T, τ).
Replacing S with S
c in Definition 6.1, we have (S
c
)
c
¼ S ¼ T À S
c . The above
definition may be rephrased as follows: Let (T, τ) be a topological space and let A be
a subset of T. Then, if A
c
¼ T À A 2 τ, then A is a closed set in (T, τ).
In parallel with the axioms (O1) to (O3), we have the following axioms related to
the closed sets of (T, τ): Let e τ be a collection of all the closed sets of (T, τ).
(C1) T 2 e τ and ∅ 2 e τ, where ∅ is an empty set.
(C2) If C 1 , C 2 , Á Á Á, C n 2 e τ, C 1 [ C 2 Á Á Á [ C n 2 e τ.
(C3) If C λ λ 2 Λ
ð
Þ2e τ, \ λ2Λ C λ 2 e τ.
6.1 Set and Topology
185