Download Introduction to Topology

Introduction to Topology
Tomoo Matsumura
August 23, 2010
Contents
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Topological spaces
1.1 Basis of a Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Comparing Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Topological spaces
A topology is a geometric structure defined on a set. Basically it is given by declaring which subsets are “open” sets. Thus
the axioms are the abstraction of the properties that open sets have.
Definition 1.1 (§12 [Mun]). A topology on a set X is a collection T of subsets of X such that
(T1) φ and X are in T ;
(T2) Any union of subsets in T is in T ;
(T3) The finite intersection of subsets in T is in T .
A set X with a topology T is called a topological space. An element of T is called an open set.
Example 1.2. Example 1, 2, 3 on page 76,77 of [Mun]
Example 1.3. Let X be a set.
• (Discrete topology) The topology defined by T := P(X) is called the discrete topology on X.
• (Finite complement topology) Define T to be the collection of all subsets U of X such that X − U either
is finite or is all of X. Then T defines a topology on X, called finite complement topology of X.
1.1
Basis of a Topology
Once we define a structure on a set, often we try to understand what the minimum data you need to specify the structure. In
many cases, this minimum data is called a basis and we say that the basis generate the structure. The notion of a basis of the
structure will help us to describe examples more systematically.
Definition 1.4 (§13 [Mun]). Let X be a set. A basis of a topology on X is a collection B of subsets in X such
that
(B1) For every x ∈ X, there is an element B in B such that x ∈ U.
(B2) If x ∈ B1 ∩ B2 where B1 , B2 are in B, then there is B3 in B such that x ∈ B3 ⊂ B1 ∩ B2 .
Lemma 1.5 (Generating of a topology). Let B be a basis of a topology on X. Define TB to be the collection of
subsets U ⊂ X satisfting
(G1) For every x ∈ U, there is B ∈ B such that x ∈ B ⊂ U.
Then TB defines a topology on X. Here we assume that ∅ trivially satisfies the condition, so that ∅ ∈ TB .
Proof. We need to check the three axioms:
(T1) ∅ ∈ TB as we assumed. X ∈ TB by (B1).
(T2) Consider a collection of subsets Uα ∈ TB , α ∈ J. We need to show
[
U :=
Uα ∈ TB .
α∈J
By the definition of the union, for each x ∈ U, there is Uα such that x ∈ Uα . Since Uα ∈ TB , there is B ∈ B
such that x ∈ B ⊂ Uα . Since Uα ⊂ U, we found B ∈ B such that x ∈ B ⊂ U. Thus U ∈ TB .
(T3) Consider a finite number of subsets U1 , · · · , Un ∈ TB . We need to show that
U :=
n
\
i=1
2
Ui
∈ TB .
– Let’s just check for two subsets U1 , U2 first. For each x ∈ U1 ∩ U2 , there are B1 , B2 ∈ B such that
x ∈ B1 ⊂ U1 and x ∈ B2 ⊂ U2 . This is because U1 , U2 ∈ TB and x ∈ U1 , x ∈ U2 . By (B2), there is
B3 ∈ B such that x ∈ B3 ⊂ B1 ∩ B2 . Now we found B3 ∈ B such that x ∈ B3 ⊂ U.
– We can generalize the above proof to n subsets, but let’s use induction to prove it. This is going to
be the induction on the number of subsets.
∗ When n = 1, the claim is trivial.
∗ Suppose that the claim is true when we have n − 1 subsets, i.e. U1 ∩ · · · ∩ Un−1 ∈ TB . Since
U = U1 ∩ · · · ∩ Un = (U1 ∩ · · · ∩ Un−1 ) ∩ Un
and regarding U 0 := U1 ∩ · · · ∩ Un−1 , we have two subsets case U = U 0 ∩ Un . By the first
arguments, U ∈ TB .
Definition 1.6. TB is called the topology generated by a basis B. On the other hand, if (X, T ) is a topological
space and B is a basis of a topology such that TB = T , then we say B is a basis of T . Note that T itself is a
basis of the topology T . So there is always a basis for a given topology.
Example 1.7.
• (Standard Topology of R) Let R be the set of all real numbers. Let B be the collection of all open
intervals:
(a, b) := {x ∈ R | a < x < b}.
Then B is a basis of a topology and the topology generated by B is called the standard topology of R.
• Let R2 be the set of all ordered pairs of real numbers, i.e. R2 := R × R (cartesian product). Let B be the
collection of cartesian product of open intervals, (a, b) × (c, d). Then B is a basis of a topology and the
topology generated by B is called the standard topology of R2 .
• (Lower limit topology of R) Consider the collection B of subsets in R:

(
)






B := [a, b) := {x ∈ R | a ≤ x < b} 
a, b ∈ R .


This is a basis for a topology on R. This topology is called the lower limit topology.
The following two lemmata are useful to determine whehter a collection B of open sets in T is a basis for T or
not.
Lemma 1.8 (13.1 [Mun]). Let (X, T ) be a topological space. A collection B of open sets is a basis for T if and
if any open set is a union of open subsets in B.
Lemma 1.9 (13.2 [Mun]). Let (X, T ) be a topological space. A collection B of open sets is a basis for T if and
if for every U ∈ T and x ∈ U, there is B ∈ B such that x ∈ B ⊂ U.
1.2
Comparing Topologies
Definition 1.10. Let T , T 0 be tow topologies for a set X. T 0 is finer than T or T is coarser than T 0 if T ⊂ T 0 .
The intuition for this notion is “(X, T 0 ) has more open subsets to separate two points in X than (X, T )”.
Lemma 1.11 (13.3). Let B, B0 be bases of topologies T , T 0 on X respectively. Then T 0 is finer than T ⇔
∀x ∈ X, ∀B ∈ B s.t. x ∈ B, ∃B0 ∈ B0 s.t. x ∈ B0 ⊂ B.
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References
[Mun] Muncres, Topology, 2nd edition
[Set]
Basic Set Theory, http://www.math.cornell.edu/∼matsumura/math4530/basic set theory.pdf
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