As the title states, I am trying to prove that the geometric realization of a simplicial set $X : \Delta^{op} \to \mathsf{Set}$ is always a CW complex. I apologize in advance for asking more than one question, but these all seem to belong to one 'package'.
Since my intention is to include this in a final presentation for a course I took which didn't require much category theory, I am looking for the less nonesense-relying argument I can find. The references I am working with are book of May, Simplicial Objects in Category Theory, and Goerss and Jardine's Simplicial Homotopy Theory.
In the first, the author claims that the result can be easily proven from the fact that the pairs of non-degenerate simplices and interior points form a system of representatives of the geometric realization.
I have proved that each pair $(x,p)$ with $x \in X_n$ and $p \in \Delta^n$ is equivalent to a unique pair $(y,q)$ with $y$ non-degenerate and $q$ interior, but I am failing to see how this implies that the geometric realization is a CW complex. I do realize that a cell has to be the projection of $\{x\} \times \Delta^n$ with $x \in X_n$ non-degenerate.
In the second book, the authors claim that $X$ is the 'union' of its skeletons (I assume this is to be understood as a colimit of 'nested' subcomplexes) and each skeleton is obtained (as a pushout) by adjoining non degenerate simplices to a previous skeleton (MathJax seems not to support tikzcd, the diagram can be found at the end of page 8 here). Since $| \cdot |$ is a left adjoint, it preserves colimits, and so the former shows that $|X|$ is the colimit of the realizations of each skeleton, which are obtained adjoining simplices (in $\mathsf{Top}$).
Here I am failing to find a non-categorical definition of skeleton, all I have found involves left-adjoints or Kan extensions. What is an elementary definition of $\mathsf{sk}_nX$? Even then, it is not clear to me why should't we also prove that $|X|$ has the final topology with respect to each skeleton. Is this a consecuence from the fact that this space is a colimit of the 'inclusion' of skeletons?
I would really appreciate if you could explain any of these proofs with some more detail than the presentations referenced above. I 'know' why this should be true, but I am struggling to fill in the details, and similar questions in this site do not seem to address my questions.
An elementary definition of $\mathrm{sk}_n X$ is: the smallest simplicial subset of $X$ containing all nondegenerate simplices of dimension at most $n$. The $n$-skeleton is obtained from the $(n-1)$-skeleton by pushing out all the nondegenerate $n$-simplices along the canonical map from their boundary into the $(n-1)$-skeleton, and this pushout process defines the attaching maps for the $n$-skeleton of the CW-complex $|X|$.
We certainly should still prove that $|X|$ has the largest topology making all the inclusions $|X_n|\to |X|$ continuous. However, this is a fact about colimits of spaces, not about geometric realization. By the definition of the colimit, a map out of $|X|$ is precisely a consistent family of continuous maps from each $|X_n|$. In particular, a map out of $|X|$ is continuous if and only if its restrictions to each $|X_n|$ are. Even more particularly, a map $|X|\to \text{Sr}$, where $\text{Sr}$ is the Sierpinski space, is continuous if and only if each restriction to $|X_n|$ is. But maps into Sierpinski space are naturally identified with the topology on $|X|$, so we find that a subset of $|X|$ is open if and only if each restriction to $|X_n|$ is, as desired.