By V. I ArnolÊ¹d

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This follows, however, from the inequality Id(x,y) - d(x,y' 1 I + Jd(x,y') - d(x' ,Y') I Id(x,y) - d(x' ,y') I -< d(y,y') + d(x,x'). 17, as R is complete, has a continuous extension d" to the completion G"XG" of GXG. Thus d*-'( [O,+m)) is a closed set containing the dense subset G X G of G"XGA and hence is all of G"XG", so d"(x,y) -> 0 for all x, ~ E G " . Also f: (x,y> -> d"(x,y) - d"(y,x) is a closed is continuous from G"XG" to R; therefore f-'(O) set containing G X G and hence is all of G"XG", so d"(x,y) = d"(y,x) for all x, ~ E G " .

1) G/II is a locally compact group. (2) C is the intersection of all the open subgroups of G. (3) G / I I is totally disconnected if and only if I€ 1 C. ( 4 ) G is connected if and only if every neighborhood of e generates 22 Topological Groups Proof. 9. 11, the intersection of all the open subgroups of G / C is {cpc(e)}. But if L is an open subgroup of G / C , cp-l(L> is an open C subgroup o f G . 3, identical with cpi1(cpc(e)) = C . Therefore (2) holds. (3) If I€ does not contain C , cp,,(C) is a connected subset of G / I I containing more than one point.

Let II be a normal subgroup of a topological group G contained in the kernel K of a homomorphism be f from G to a topological group G I , and let g: G / I I - > G I the homomorphism satisfying goqI = f. Then g is continuous [open, a topological homomorphism] if and only if f is. In particular, if I€ = K , g is a topological isomorphism [monomorphism] if and only if f is a topological epimorphism [homomorphism]. 5. 14. Theorem. Let II be a normal subgroup and A a subgroup of a topological group G .