# CW Complex: top dimension cells

1. Apr 11, 2010

### arithmuggle

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 :
$$\phi: (D^k, S^{k-1}) \rightarrow (e \cup X^{k-1}, X^{k-1})$$

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

4. A subset $$A \subset X$$ is closed if and only if $$A \cap \overline{e}$$ is closed in $$\overline{e}$$ 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