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Homework Help: CW Complex: top dimension cells

  1. Apr 11, 2010 #1
    1. The problem statement
    Given an n-dimensional CW complex on a space X, show that each n-cell is open in X.

    Definition of CW Complex
    1. The cells form a disjoint union of X
    2. For each k-cell e, there is a relative homeomorphism :
    [tex]\phi: (D^k, S^{k-1}) \rightarrow (e \cup X^{k-1}, X^{k-1})[/tex]

    3. For each k-cell e, [tex] \phi(S^{k-1}) \subset \bigcup\limits_{i=1}^n e_i[/tex] where each e_i has dimension less than k.

    4. A subset [tex]A \subset X [/tex] is closed if and only if [tex]A \cap \overline{e} [/tex] is closed in [tex] \overline{e}[/tex] for every e.

    Solution work
    I feel that I can show that the n-1 skeleton is closed and so showing that the n-cells are components would give it.


    Otherwise I was just trying to play with the weak topology condition to see if I can show it directly.

    Either way I'm having trouble.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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