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

12 Topology (2301631)
Proof. (⇒) Trivial.
(⇐) Let U ∈< B > and x ∈ U. Then there is B ∈ B such that x ∈ B ⊆ U. By
assumption, we can find B ′ ∈ B ′ such that x ∈ B ′ ⊆ B. Therefore, x ∈ B ′ ⊆ U;
i.e., U ∈< B ′ >.
□
Corollary 1.28. Let (X, τ) be a space and a basis B such that B ⊆ τ. Then
τ is generated by B iff for each U ∈ τ and x ∈ U, there exists B ∈ B such that
x ∈ B ⊆ U.
Example 1.29. The usual topology on R n is generated by the collection of
open balls {B(x; r) | x ∈ R n , r > 0}. In particular, the usual topology on R is
generated by {(a, b) | a < b}.
Definition 1.30. The lower limit topology on R is the topology generated by
{[a, b)|a < b}. We simply write R l to denote R equipped with the lower limit
topology.
Exercise 1.31. Prove that the lower limit topology on R is strictly finer than
the usual topology.
Definition 1.32. Let (X, ≤) be a totally ordered set with |X| > 1. An interval
in X is one of the following sets :
(a, b) = {x ∈ X | a < x < b}
[a, b) = {x ∈ X | a ≤ x < b}
(a, b] = {x ∈ X | a < x ≤ b}
[a, b] = {x ∈ X | a ≤ x ≤ b}
The order topology on X is the topology generated by the basis consisting of all
intervals of the following forms :
• (a, b),
• [x min , b) if X has the smallest element x min ,
• (a, x max ] if X has the largest element x max .
Exercise 1.33. Consider the set R 2 = R × R whose element is denoted by
a × b, and the dictionary order (≼) on R 2 defined by
a 1 × b 1 ≼ a 2 × b 2 iff (a 1 × b 1 = a 2 × b 2 ) or (a 1 × b 1 ≺ a 2 × b 2 ),
where a 1 × b 1 ≺ a 2 × b 2 iff a 1 < a 2 or (a 1 = a 2 and b 1 < b 2 ).
Is the dictionary order topology on R × R the same as the usual topology on
R 2 ? If not, compare them.