Covering of a CW-complex is a CW-complex

Question

Let $X$ be a CW- complex, with filtration $\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X$. Let $p\colon E \to X$ be a covering space. Prove that $E$ is a CW complex with filtration given by the family $E_i := p^{-1}(X_i)$.

First recall the definition:

Definition.A CW complex $X$ is a space $X$ which is the union of an expanding sequence of subspaces Xn such that, inductively, $X_0$ is a discrete set of points (called vertices) and $X_{n+1}$ is the pushout obtained from $X_n$ by attaching disks $D^{n+1}$ along “attaching maps” $j\colon S^n \to X_n$

Call the characteristic maps for $X$ in the usual way: $Q_i^n \colon D_i^n \to X_n$ where $i$ is only an index.

My attempt:

I lifted my $Q_i^n$ and obtained a family $\{\tilde{Q^n}_{i,j}\}$ of maps with $Q_i^n = p\circ \tilde{Q^n}_{i,j} $ for all $j \in J$ (I'm not assuming that the degree of our covering is finite. I managed to prove that the topology on $E$ is the weak topology, but I'm not able to prove that it is a push out square with these characteristic maps (and their restriction to $S^{n-1}$). I've searched around and it should be easy (because lots of books says so , e.g. here and here, or give it as an exercise as Hatcher's pg. 529)

I've proved that for every $n$, $$E_n = E_{n-1} \ \ \bigcup \ \ \left(\bigcup_{i,j}\tilde{Q^n}_{i,j}(D^n_{j,i}) \right)$$

But I'm unable to find an homeo between this space and the usual push out in Top which turns out to be $$E_{n-1} \bigsqcup_{\eta} \bigcup_{i,j}D_{i,j}^n $$ where $\eta$ is th projection which identifies the image of elements in $S^{n-1} \subset D^n$ with their images through the respective characteristic maps. In fact I found a continuous bijection between the two spaces, but from there I'm unable to proceed (because It's not necessarily compact, or the characteristic maps are not necessarily open)

Someone has got some advice or can show me how to prove this?

Answer 1

This is proved in Massey's "Algebraic topology" which I do not have with me.

There is a more general result. We work in the category $\mathsf{ Top} $ of compactly generated spaces ($=$ weakly Hausdorff k-spaces}. Let $p:Y \to X$ be a map in this category. Then the pullback functor $p^*:\mathsf{ Top}/X \to \mathsf{ Top}/Y$ has a right adjoint, and so preserves colimits. This argument is used in Section 4 of this paper, which gives further references, including this paper on partial maps.

The point is that if $p$ is a covering map and $X$ is a CW-complex then the possibly relevant cell structure on $Y$ is easily described, but the problem is one of the correct topology. That it is the colimit topology is shown by the above argument.