Cell Complex: Proposition 5.5 in John Lee's book "Introduction to Topological Manifolds" The proposition reads:
"Suppose $X$ is an $n$-dimensional CW complex. Then every $n$-cell of $X$ is an open subset of $X$."
The proof first shows that the intersection of any $n$-cell of $X$ with the closure of any other cell is the empty set. From this it is concluded that the intersection of any $n$-cell with the closure of every cell is open. Then it follows from the (W)-condition that the $n$-cell is open.
So far I can follow. However, the empty set is also closed. Thus, in my opinion, the conclusion could also be that every $n$-cell of $X$ is a closed subset of $X$. What am I missing?
 A: I don't have access to Lee's book, but usually we have characteristic maps $\Phi_\alpha:D^k_α\to X$ from the unit ball $D^k$ (where $k$ depends on the dimension of the cell $α$) to the CW complex $X$. The cell $e_α$ is then defined as the image $Φ_α(\text{int}D_α^k)$ of the interior. Using the Hausdorffness of $X$, it follows that the closure $\overline {e_α}$ of a cell is the image $Φ_α(D_α^k)$. Saying that $A\cap\overline{e_α}$ is closed is then the same as $Φ_α^{-1}(A)$ being closed in $D^k$.
Anyway, since the $n$-cell $e_\beta$ is disjoint from $Φ_α(D^k_α)$ for every $α\ne\beta$ (here it is important that there are no cell of higher dimension), these intersections are indeed empty, so it suffices to consider $Φ_\beta^{-1}(e_\beta)$, but that is just $\text{int}(D^n_\beta)$ as no point in $\partial D^n$ is mapped to $e_α$. So every preimage is open, making $e_α$ open.
A: There is a general result here (see Topology and Groupoids p.121). 
A useful construction, due to JHC Whitehead, is that of the adjunction space $B \cup_f X$ from a space $B$ and a map $f: A \to B$ from a closed subset $A$ of $X$. We now recognise this as a special kind of pushout 
$$\begin{matrix} A & \xrightarrow{f} & B \\
\downarrow && \downarrow\\
X & \to & B \cup_f X \end{matrix}$$
These conditions imply that the map $X \backslash A \to B \cup_f X$ is an open map. 
