By Ol'shanskii A. Y.

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**Example text**

I - $’ vanishes on all p i bi, hence on all b E @ B i , that is, $ - $’ = 0 . 2. Let c$i: A + B , be homomorphisms, i unique homomorphism $ making all the diagrams commute; here x i denote the projections. E I. There exists a 8. 41 DIRECT SUMS AND DIRECT PRODUCTS I7 For a E A , define $a as the unique b E Bi with n i b = 4 i a [cf. (b”)]. This $ is evidently a homomorphism such that r i $ = 4 i for every i. It is unique, for if $’ is a homomorphism with the same stated property, then xi($ - $’)a = 0 for every i, and thus ($ - $’)a E Bi has only 0 coordinates.

Assume A topological in the closed 6-topology. Given an open set V = A\(a T ) (a E A , T E 6 ) containing 0, there is an open neighborhood U of 0, such that U - U E V. We may write + u = A\ u n i= 1 + S,) (Ui with a, E A , S, E 6, since every open set is the union of open sets of this form. Since U - U E V implies for every u E U , u - a $ U , therefore u E u:=,(a a, Si), and + + + A = u (a, + Si) u u ( a + a , + Si). 3), 1 m 5 n. Let S = S, n .. n S, which is again of finite index in A . We claim S E V.

A*p. I , and njapi=O =nJa*p, (i#j) for the injections p i of the A i and projections n: of the B i . Namely, a [a*] sends the ith coordinate a, upon the ith coordinate a i a i . We shall denote them as a =@ai and a* = n a i . i i For a group G, we introduce two maps: the diagonal map AG : G [the number of components can be arbitrary] as A,: g - ( . . ) -+ n G ( 9 E GI, , and the codiagonal map V , : 0 G + G as V G : ( . * , g i , * * * ) ~ C (ggii E G ) . i If there is no danger of confusion, we may suppress the index G.