# Borel Equivalence Relations: Structure and Classification by Vladimir Kanovei ###############################################################################################################################################################################################################################################################

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1. Thus, the " g e o m e t r i c " description of a locally m-convex algebra given in Section 1 , will be supplemented by an " a n a l y t i c " one. However, we need first the following easy, but useful, result. Lemma 3 . 3 . Let E be a t o p o l o g i c a l aZgebra and U an m-barrel of E which i s ,, neighborhood of z e r o . Moreover, l e t p u be t h e r e s p e c t i v e gauge f u n c t i o n of U. 12) u = I X E B : p U (x-i 6 1 1 ; U c o i n c i d e s w i t h t h e r e s p e c t i v e c l o s e d u n i t semi-ball of its gauge f u n c t i o n .

In fact, something more holds true on the basis of the following result from the general theory of topological vector spaces. I 24 GENERAL CONCEPTS Lemma 4 . 2 . Let E,F, G be topological vector spaces such t h a t E i s a Baire space and F metrizable. ~~c:raover, the same holds t r u e i f E i s a barrelled rnetrizable (topological v e c t o r ) space and G any l o c a l l y convex space. Proof. See H. SCHAEFER [l: p. 881 and F . TREVES [l: p. 4251. 1. Every Baire metrizable topological algebra has (jointly) continuous r m l t i p i i c a t i o n .

Let E be a t o p o l o g i c a l aZgebra and U an m-barrel of E which i s ,, neighborhood of z e r o . Moreover, l e t p u be t h e r e s p e c t i v e gauge f u n c t i o n of U. 12) u = I X E B : p U (x-i 6 1 1 ; U c o i n c i d e s w i t h t h e r e s p e c t i v e c l o s e d u n i t semi-ball of its gauge f u n c t i o n . 13) U = { x& E :pU(xl < 1 1. 12). Proof. 5. m I GENERAL CONCEPTS 18 Thus, we now have the following. 2. L e t E be a l o c a l l y m-convex aZgebra (Definition 1 . 1 ) Then , there e x i s t s a fundamental defining family (of submultiplicative semi-norms)of the topology o f E.