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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers according to talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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This implies there are E > 0 and N 2 1 such that n,, - W, = A. Thus We now define a new sequence of open symmetric neighborhoods of A by vo = wo, vn = Wl+(*-l)N, Obviously, n 2 1. V, = A. 11 we must prove that V,+l o Vn+l o V,+l for n 2 0. c V, It is clear that V1 o V1 o V1 c VOand by the choice of E and N , V2 o Vz o VZ C V1. Let k 2 2 and take (z,y) E vk+1Ovk+1Ovk+1. Then we prove that (z,y)E v k . Let z , w E X be points such that (z,w), (w,z),(z,y) E Vk+1. If ( p , q ) is any of these three points, then we have from which d(fi(z),fi(y)) < e, lil < 1 + (k - l)N.

Here d(A,B) is defined by d ( A , B )= inf{d(a,b) : a E A,b E B}. To show (4) let 1 > 0. For 1 5 i 5 m there is 0 < ~i < X such that if d(f(z),f(z)) < ~i (f(z), f(z) E f(cl(Ui))) then d(z,z ) < 7. This follows from the fact that f:cl(Ui) --f f(cl(Ui)) is injective. Put E = min{si : 1 5 i 5 m } . If diam(D) < E , then we have diam(Di) < 1 for 1 5 i 5 k. e. f is k-to-one. - . (n;=, u,"==, In the remainder of this section we describe well known theorems that will be used in the sequel. Let X be a topological space.

If this is false, for any n > 0 there exists a subset A, such that B for all B E a. diam(A,) = sup{d(a,b) : a , b E A,} 5 l / n and A, Take x , E A,, for n 2 1. Since X is compact, there is a subsequence {x,,} such that x,, 4 x as i -+ 00. Then x E B for some B E a. Since X \ B is compact, we have a = d ( x , X \ B ) = inf{d(x,b) : b E X \ B} > 0. Take sufficiently large ni satisfying ni > 2 / a and d(x,, ,x ) < a/2. Then, for y E Ani we have d ( y , ~ ) I d ( ~ i x n i ) + d ( z n i , l~/ n ) Ii + a / 2 < a < and thus y E B.