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) = jθ1 − θ2j. Then form the quotient R T := = f[θ]: θ 2 Rg ; 2πZ 1 where a coset [θ] is [θ] := fθ + 2πk 2 R: k 2 Zg. 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 f[θ]: θ 2 Ug where U is open in R. This quotient topology τT is induced by the quotient metric dT ([θ1] ; [θ2]) := infk2Z fjθ1 − θ2 + 2πkjg. 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] 2 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) = jz1 − z2j. Let iθ iθ S := fz 2 C: jzj = 1g = e 2 C: θ 2 R = e 2 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; 1) 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 (f[θ]: θ 2 Ug) = eiθ : θ 2 U : Furthermore, for all θi 2 R, iθ1 iθ2 dT ([θ1] ; [θ2]) := inf fjθ1 − θ2 + 2πkjg = dS e ; e = ds (I ([θ1]) ;I ([θ2])) : k2Z 1 Thus [θ1] = [θ2] if and only if θ1 − θ2 2 2πZ. 2A topological space X is sequentially compact provided each sequence in X has a convergent subsequence. 2014 November 19 Page 1 of 1.