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 1 where a coset [θ] is [θ] := {θ + 2πk ∈ R: k ∈ Z}. 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]) := infk∈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. 2 T is also compact since compactness is equivalent to sequential compactness 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 iθ iθ S := {z ∈ C: |z| = 1} = e ∈ C: θ ∈ R = e ∈ 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 iθ I :(T, +, τT) → (S, ·, τS) given by I ([θ]) = e .

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,

iθ1 iθ2 dT ([θ1] , [θ2]) := inf {|θ1 − θ2 + 2πk|} = dS e , e = ds (I ([θ1]) ,I ([θ2])) . k∈Z

1 Thus [θ1] = [θ2] if and only if θ1 − θ2 ∈ 2πZ. 2A topological space X is sequentially compact provided each sequence in X has a convergent subsequence.

2014 November 19 Page 1 of 1