Download Prof. Girardi The Circle Group T Definition of Topological Group A

Prof. Girardi
The Circle Group T
Definition of Topological Group
A triple (G, ∗, τ ) is a topological group provided G is a set that is so endowed that
• (G, ∗) is an group in the algebraic sense (with binary operation ∗ : G → G)
• (G, τ ) is a topological space
and these two structures interact so that the group operations are continuous with repect to τ ,
i.e., the mappings
• (G × G, product topology induced by τ ) → (G, τ ) given by (θ1 , θ2 ) 7→ θ1 ∗ θ2
• (G, τ ) → (G, τ ) given by θ 7→ the inverse of θ
are continuous.
Definition of T
T is commonly called the circle group or the one dimensional torus group. To define T, start by
viewing R as a topological group, endowed with the group binary operation of addition and with
the topology induced by the metric dR (θ1 , θ2 ) = |θ1 − θ2 |. Then form the quotient
R
T :=
= {[θ] : θ ∈ R} ,
2πZ
where a coset [θ] is [θ] := {θ + 2πk ∈ R : k ∈ Z}.1 Then T is a topological group with the inhertited
quotient structure. Specifically, the group structure of T is
[θ1 ] + [θ2 ] = [θ1 + θ2 ]
and
− [θ] = [−θ] .
The collection of open subsets of T, i.e., the quotient topology on T, precisely consists of all sets of
the form {[θ] : θ ∈ U } where U is open in R. This quotient topology τT is induced by the quotient
metric dT ([θ1 ] , [θ2 ]) := inf k∈Z {|θ1 − θ2 + 2πk|}.
Let’s look at some nice properties of T. Consider the natural projection π : R T given by
π (θ) = [θ]. Then π is continuous since if dR (xn , x) → 0 then dT ([xn ] , [x]) → 0. Following directly
from the definition of the quotient topology is that π is an open mapping and that T is Hausdorff.
T is also compact since compactness is equivalent to sequential compactness2 in a metric space
and for any [θ] ∈ T there exists θ0 in the compact set [0, 2π] so that [θ] = [θ0 ].
There is another fruitful way to view T. Endow C with its usual topology given by the metric
dC (z1 , z2 ) = |z1 − z2 |. Let
S := {z ∈ C : |z| = 1} = eiθ ∈ C : θ ∈ R = eiθ ∈ C : 0 ≤ θ < 2π ,
be endowed with the subspace topology τS from C. Note that τS is generated by the metric
dS : S × S → [0, ∞) where dS (z1 , z2 ) is the arclength of the shortest arc along S from z1 to z2 .
Then (S, ·, τS ) is a topological group. Consider the mapping
I : (T, +, τT ) → (S, ·, τS )
I ([θ]) = eiθ .
given by
Then I is a group isomorphism, i.e, I is bijective and I ([θ1 ] + [θ2 ]) = I ([θ1 ]) · I ([θ1 ]). I also
preserves the topology (i.e V is open in T if and only if I (V ) is open in S) since
I ({[θ] : θ ∈ U }) = eiθ : θ ∈ U .
Furthermore, for all θi ∈ R,
dT ([θ1 ] , [θ2 ]) := inf {|θ1 − θ2 + 2πk|} = dS eiθ1 , eiθ2
k∈Z
1Thus
2A
= ds (I ([θ1 ]) , I ([θ2 ])) .
[θ1 ] = [θ2 ] if and only if θ1 − θ2 ∈ 2πZ.
topological space X is sequentially compact provided each sequence in X has a convergent subsequence.
2014 November 19
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