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By Boyer Ch. P.

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In particular, a IIausdorff group is a completely regular topological space. Proof. If hl has the desired properties for e and a-lF, then h: x->h1(a-'x) has the desired properties for a and F. Therefore we may assume that a = e. 2. n 2 0 . Let f be the function associated to (Un)n, By 8" and 9 " of that theorem, if y - l x ~ u k ,then If(x)- f(y)l -< 2-k, so f is left uniformly continuous from G to the additive topological group R. If X E F, then X E G\Uo, so by 1 has the 9 " , f(x) > 7 . Consequently, h: x->inf{2f(x),l} desired properties..

If HI, I In ~ n 11, = I I n ~ nq,. 3. + ... . . 6. aLed to ( x ~ >)1,~ having as base the sets Sm for m i l , where S, = {x,: n i m If G is a topological space, the sequence conweage6 to a E G if the associated filter does, that is, if for every neighborhood V of a there exists m such that X,E V for all n i m . If G is a topological group, we say that (xnIn > 1 is a left [right, bilateral] Cauchy dequmce if the associated filter is a left [right, bilateral] Cauchy filter. For example, ( x ~ ) ~l >is a bilateral Cauchy sequence if and only if for every-neighborhood V of e there exists m such that G 1 x c V and x n S 1 E V for all n i m , p,m.

Finally, assume that each Uk is a < g(x) for all subgroup. We have already seen that f(x) X E G and f(x) = g(x) = 0 for all X E A{Un: n E Z } . Assume that x E Un\Un+l. If (zi>l< L p is any sequence such that 2122. ) > 2-("+l) and hence g(zj) J P -> 2-", and consequently c g ( z i ) -> 2-n = g(x) by 3 " . * 6 . 3 . Theorem. If F is a closed subset of a topological group G and if aEG\F, there is a continuous function h from G into [0,1] such that h(a) = 0 and h(x) = 1 for all X E F. In particular, a IIausdorff group is a completely regular topological space.

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3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients by Boyer Ch. P.


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