I've the following definition
a (finite) CW-complex of dimension $N$ is a topological space $X$ defined in the following way:
- $X^0$ is a discrete set of points
- $\forall 0<n\le N$, $X^n$ is obtained from $X^{n-1}$ by attaching a finite number $J_n$ of closed n-balls $\{B^n_j\}$ via maps $\phi^n_j: S^{n-1}_j \to X^{n-1}$; in symbols $X^n=[X^{n-1} \cup (\sqcup_{j=1}^{J_n} B^n_j)] / \sim$, where $\sim$ is the equivalence relation generated by $x \sim y$ if $\exists j \in \{1,...,J_n\}: y=\phi^n_j(x)$
So for every $j$ I have the following characteristic map given by the composition
$\Phi^n_j:B^n_j \longrightarrow X^{n-1} \cup (\sqcup_{j=1}^{J_n} B^n_j) \longrightarrow X^n \longrightarrow X $
where the first and the last arrow are inclusions and the one in the middle is the quotient projection $\pi^n$
Now, $X^n$ is a quotient of $X_{n-1}\cup ...$ so it isn't true that $X^{n-1} \subseteq X^n$. So I have to identify $X^{n-1}$ and its image under $\pi^n$ to make the last arrow an inclusion; all this for every $n$.
How can I prove that $X^{n-1} \simeq \pi^n(X^{n-1})$?
And then, how can I show that for every $n$ $X^n$ is an Hausdorff compact space?