By M.M. Cohen

Cohen M.M. A direction in simple-homotopy idea (Springer, [1973)(ISBN 3540900551)

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3 (to follow) is a slight extension of the degree developed by Browder and Petryshyn [16a,b]. 1. t. r and yE Y is such that y is T(aGD). Then there exists No E Z+ such that Qn y e Tn (aGn) for alln>_No. Proof. 1 is false. Then there exists a sequence [ nj] C Zl with nj - oo as j - co and a sequence xni E Gni such that Tni (xn) = Q,, y. Since Q,J(y) - y and Tn, (x,,) -y, the A-properness of T implies the existence of a subsequence Ix y] and x'E GD such that x;,j - x' and T(x') = y. Since x' E GD = G fl D, it follows that X' E 0 and X E D.

For subsequent use we will re- call that T: X- Y is said to be pseudo-A-proper if T,: X Y, is continuE X,y1 is bounded and 0 in Y as j -+ 00 ous and, if I for some g in Y, then there exists an x C X such that T(x) = g. As is well known (see [66e; 73c,r]) and as we shall demonstrate, this is a very large class of mappings. 1. 3 can be found in Heinz [42] or Deimling [24b]. 1 can be found in [24b] or Lloyd [59]. 2. 10 can be found in Schwartz [S], Lloyd [59], and Deimling [24b]. 3. For the reader who is interested in more general approximation schemes within the framework of A-proper mapping theory, where and {Yn] are not subspaces of X and Y (respectively) and and are not projections, see Petryshyn [73m], Chapters 34-36 in [112b], and Ruotsalainen [S].

R and yE Y is such that y is T(aGD). Then there exists No E Z+ such that Qn y e Tn (aGn) for alln>_No. Proof. 1 is false. Then there exists a sequence [ nj] C Zl with nj - oo as j - co and a sequence xni E Gni such that Tni (xn) = Q,, y. Since Q,J(y) - y and Tn, (x,,) -y, the A-properness of T implies the existence of a subsequence Ix y] and x'E GD such that x;,j - x' and T(x') = y. Since x' E GD = G fl D, it follows that X' E 0 and X E D. But xni E aGn C aG and hence x'e aG and x'E D. Thus xE aGflD = aGD, in contradiction to the assumption that y is T(aGD).