CW Complex: top dimension cells

[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.

 

[lippyka]

Any help or ideas would be greatly appreciated. 2. Solution Let X be an n-dimensional CW complex. We will show that each n-cell is open in X. By definition of a CW complex, for each n-cell e, there is a relative homeomorphism: \phi: (D^n, S^{n-1}) \rightarrow (e \cup X^{n-1}, X^{n-1}) Note that the domain of the homeomorphism is open in the Euclidean space, so the image of this map must also be open in X. Since the image of the homeomorphism is the n-cell e, it follows that e must be open in X. This completes the proof.