Lecture Notes Topology (2301631) Phichet Chaoha Department of ...

6 Topology (2301631)
Proof. Let {A α } α∈Λ be a countable collection of countable sets. Then there
are surjections i : N → Λ and f α : N → A α for each α ∈ Λ. It follows that the
function F : N × N → ∪ α∈Λ A α defined by
is a surjection.
F (m, n) = f i(m) (n)
Example 0.6. Z × Z and Q are countable.
Theorem 0.7. A finite product of countable sets is countable.
Example 0.8. If B i is a countable set for each i ∈ N, the set ∪ n∈N
countable.
Ordered Sets
□
n∏
B i is also
Definition 0.9. A relation ≤ on a set A is called a preorder if it is reflexive
and transitive. When ≤ is a preorder on A, we will call (A, ≤) a preordered set.
Definition 0.10. A relation ≤ on a set A is called a partial order if it is
reflexive, antisymmetric and transitive. When ≤ is a partial order on A, we will
call (A, ≤) a partially ordered set or a poset.
It is not difficult to see that a preorder ≤ on a set A induces a partial order on
A/ ∼ , where ∼ is the equivalence relation on A defined by a ∼ b iff a ≤ b and b ≤ a.
Theorem 0.11 (Zorn’s lemma). Let (A, ≤) be a nonempty partially ordered
set. If every chain in A has an upper bound in A, then A has a maximal element.
Definition 0.12. A partially ordered set (A, ≤) is said to be totally ordered if
every a, b ∈ A, we have a ≤ b or b ≤ a.
Definition 0.13. A totally ordered set (A, ≤) is said to be well ordered if every
nonempty subset of A has a smallest element.
Definition 0.14. For a well ordered set (A, ≤) and α ∈ A, the section of A by
α defined to be the set S α = {x ∈ A | x < α}.
Theorem 0.15. There exists a well ordered set L having the largest element
Ω such that the section S Ω is uncountable but every other section is countable.
Moreover, each countable subset of S Ω has an upper bound in S Ω .
i=1