Download A Textbook of Topology; Topology of 3-Dimensional Fibered by Herbert Seifert, William Threlfall PDF

By Herbert Seifert, William Threlfall

ISBN-10: 0126348502

ISBN-13: 9780126348507

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Extra resources for A Textbook of Topology; Topology of 3-Dimensional Fibered Spaces

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On pose a = lim xn . +1 est un compact de E . Posons x0 = a , on a ainsi A = xn / n ∈ Si . d’e´le´ments de A est de´finie par une application ; est borne´e, on peut extraire de x ! : telle que : , un = x (n) . n∈ ; Si est non borne´e, il existe une suite d’entiers ; strictement croissante de limite +1. Alors u (n) n∈ une suite constante, donc convergente. ; est extraite de x (n ) n∈ telle que la suite n (n) ;; (n ) soit et converge vers a . Ainsi toute suite de A admet au moins une valeur d’adhe´rence.

E´quivalence des normes The ´ore `me 17 Sur ☞ n deux normes quelconques sont e´quivalentes. L’e´quivalence des normes est une relation transitive, il suffit donc de comparer une norme quelconque N de n a` la norme k • k1 de n pour conclure. ; Notons ei 1 i n Tout vecteur x de n la base canonique de n X X ; N e n et = > 0. i i=1 n s’e´crivant x = X n N (x ) i=1 ce qui donne l’ine´galite´ N (x ) xi ei , on a : i=1 ; jx j N e i k x k1 i k x k1 X ; ! N e n i i=1 pour tout x de n ; . Cela prouve aussi que l’application N : !

1 ;F n ;F n puis sup kxn+p ; xn k ; p∈ = 0, il en re´sulte que xn ;F n . est une suite de Cauchy de A et elle converge donc dans A (qui est complet). Posons = lim xn . Pour tout p ∈ , on a n p, xn ∈ Fn ⊂ Fp donc = lim xn ∈ Fp . +1 Or, puisqu’elle est comple`te, la partie A est ferme´, donc Fp ferme´ de A est aussi un ferme´ de E et Fp = Fp , et finalement ∈ Fp . +1 ;F \ n∈ = 0, il vient n Fn . ;F n∈ Fn , on a ; 0 n ∈ ; 0 , = 0 et finalement \ Fn = f n , donc, puisque g. n∈ The ´ore `me 5 MP* (53) ✎ On dit que f est contractante.

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