- #1
PsychonautQQ
- 784
- 10
Gluing Lemma: Let X be a topological space, and suppose X = A_1 U A_2 U ... U A_k, which each A_i is closed in X. For each i, let f_i: A_i ---> Y be a continious map s.t f_i = f_j on the intersection of A_i and A_j.
Then the book goes on to give an example of where this is not true for an arbitrary union rather than a finite union:
By consider the subspace X=[0,1] of R, and the sets A_0 = {0} and A_i = [1/(i+1) , 1/i] for i=1,2,3... the gluing lemma is false.
The proof is left as an exercise. This is not homework so i feel no guilty posting it here, where it's more likely to be read by somebody that has some insight into the problem. Can anybody show me why the gluing lemma no longer applies?
Then the book goes on to give an example of where this is not true for an arbitrary union rather than a finite union:
By consider the subspace X=[0,1] of R, and the sets A_0 = {0} and A_i = [1/(i+1) , 1/i] for i=1,2,3... the gluing lemma is false.
The proof is left as an exercise. This is not homework so i feel no guilty posting it here, where it's more likely to be read by somebody that has some insight into the problem. Can anybody show me why the gluing lemma no longer applies?