(adsbygoogle = window.adsbygoogle || []).push({}); 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

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# Homework Help: CW Complex: top dimension cells

Can you offer guidance or do you also need help?

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