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From: achille on 24 Jul 2010 03:01 On Jul 24, 1:16 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Fri, 23 Jul 2010, hagman wrote: > > On 23 Jul., 12:35, William Elliot > > >>>> Let G be a compact group and U an open nhood of e. > >>>> Is there an open V nhood e with V subset U and > >>>> . . for all a, aVa^-1 = V? > > > The obvious candidate for V is the intersection of all aUa^-1, a in G. > > But for it to be open we better intersect only finitely many open sets > > The aU cover G, hence there is a finite subcover a_1 U, ..., a_n U ... > > You're suggesting take > . . V = a1.Ua1^-1 /\../\ a_n.U.a_n^-1 ? . . (1) > > Yes, V is an open nhood of e. Why is V a subset of U? > Do we use, > . . for all a, some open U_a nhood e with a.U_a.a^-1 subset U ? > > Then, by taking > . . V = a1.U_a1.a1^-1 /\../\ a_n.U_an.a_n, . . (2) > the desired > . . V open nhood e, V subset U. > > Yet, why for all a, is aVa^-1 a subset of V? Not only the finite intersections, the intersection of aUa^-1 for all a in G is also open!
From: William Elliot on 24 Jul 2010 04:19 On Sat, 24 Jul 2010, achille wrote: >> >>>>>> Let G be a compact group and U an open nhood of e. >>>>>> Is there an open V nhood e with V subset U and >>>>>> . . for all a, aVa^-1 = V? >> >>> The obvious candidate for V is the intersection of all aUa^-1, a in G. >>> But for it to be open we better intersect only finitely many open sets >>> The aU cover G, hence there is a finite subcover a_1 U, ..., a_n U ... >> > Not only the finite intersections, the intersection > of aUa^-1 for all a in G is also open! Clearly e in V = /\{ aUa^-1 | a in G } subset U. If v in V, a in G, then for all x, .. . some u_x in U with v = xu_x.x^-1. For all x, ava^-1 = aa^-1.x.u_(a^-1.x).(a^-1.x.)^-1.a^-1 = x.u_(a^-1.x).x^-1; for all x, ava^-1 in xUx^-1; ava^-1 in V. Thus for all a, V subset aVa^-1, QED. Ok, so far so good. Now why is V open?
From: achille on 24 Jul 2010 05:09 On Jul 24, 4:19 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Sat, 24 Jul 2010, achille wrote: > > >>>>>> Let G be a compact group and U an open nhood of e. > >>>>>> Is there an open V nhood e with V subset U and > >>>>>> . . for all a, aVa^-1 = V? > > >>> The obvious candidate for V is the intersection of all aUa^-1, a in G.. > >>> But for it to be open we better intersect only finitely many open sets > >>> The aU cover G, hence there is a finite subcover a_1 U, ..., a_n U .... > > > Not only the finite intersections, the intersection > > of aUa^-1 for all a in G is also open! > > Clearly e in V = /\{ aUa^-1 | a in G } subset U. > > If v in V, a in G, then for all x, > . . some u_x in U with v = xu_x.x^-1. > > For all x, > ava^-1 = aa^-1.x.u_(a^-1.x).(a^-1.x.)^-1.a^-1 = x.u_(a^-1.x).x^-1; > > for all x, ava^-1 in xUx^-1; ava^-1 in V. > > Thus for all a, V subset aVa^-1, QED. > > Ok, so far so good. Now why is V open? For all x in V = \intersect_{a in G} a U a^{-1}, we have forall a, x in a U a^{-1} <=> forall a, a^{-1} x a in U Since the map (a,x) -> a^{-1} x a is continuous and U open, there exists open neighbourhood A_a of a, X_a of x such that a' in A_a, x' in X_a => a'^{-1} x' a' in U <=> X_a \subset \intersect_{a' in A_a} a' U a'^{-1}. Now A_a forms an open cover of G, take a finite subcover A_{a_i}, It is then clear the set \intersect_{a_i} X_{a_i} is an open neighbourhood of x which is also a subset of \intersect_{a_i} \intersect_{a' \in A_{a_i}} a' U a'^{-1} = V Since this is true for all x in V, V is open.
From: William Elliot on 25 Jul 2010 04:18
On Sat, 24 Jul 2010, achille wrote: >> >>>>>>>> Let G be a compact group and U an open nhood of e. >>>>>>>> Is there an open V nhood e with V subset U and >>>>>>>> . . for all a, aVa^-1 = V? >> >>>>> The obvious candidate for V is the intersection of all aUa^-1, a in G. >>>>> But for it to be open we better intersect only finitely many open sets >>>>> The aU cover G, hence there is a finite subcover a_1 U, ..., a_n U ... >> >>> Not only the finite intersections, the intersection >>> of aUa^-1 for all a in G is also open! >> >> Clearly e in V = /\{ aUa^-1 | a in G } subset U. >> >> If v in V, a in G, then for all x, >> . . some u_x in U with v = xu_x.x^-1. >> >> For all x, >> ava^-1 = aa^-1.x.u_(a^-1.x).(a^-1.x.)^-1.a^-1 = x.u_(a^-1.x).x^-1; >> for all x, ava^-1 in xUx^-1; �ava^-1 in V. >> >> Thus for all a, V subset aVa^-1, QED. >> Ok, so far so good. �Now why is V open? > > For all x in V = \intersect_{a in G} a U a^{-1}, we have > forall a, x in a U a^{-1} > <=> forall a, a^{-1} x a in U > Since the map (a,x) -> a^{-1} x a is continuous and U open, > there exists open neighborhood A_a of a, X_a of x such that > a' in A_a, x' in X_a => a'^{-1} x' a' in U It never occurred to me to use this stronger version. > <=> X_a \subset \intersect_{a' in A_a} a' U a'^{-1}. > Now A_a forms an open cover of G, take a finite subcover A_{a_i}, > It is then clear the set \intersect_{a_i} X_{a_i} is an open > neighborhood of x which is also a subset of > \intersect_{a_i} \intersect_{a' \in A_{a_i}} a' U a'^{-1} = V > Since this is true for all x in V, V is open. > Whew, that was intricate. |