$n$-skeleton in the definition of CW complexes I try to understand the abstract definition of CW complexes in Hatcher's Algebraic Topology.
Specifically, I refer to the following definition in the appendix of Hatcher's book.

A CW complex is a pace $X$ constructed in the following way:


(1) Start with a discrete set $X^0$, the $0$-cells of $X$.
(2) Inductively, form the $n$-skeleton $X^n$ from $X^{n-1}$ by attaching $n$-cells $e_\alpha^n$ via maps $\varphi_\alpha:S^{n-1}\to X^{n-1}$. This means that $X^n$ is the quotient space $X^{n-1}\sqcup_\alpha D_\alpha^n$ under the identification $x\sim \varphi_\alpha(x)$ for $x\in\partial D_\alpha^n$. The cell $e_\alpha^n$ is the homeomorphic image of $D_\alpha \setminus \partial D_\alpha^n$ under the quotient map.
(3) $X=\cup_n X^n$ with the weak topology.

I do not understand the inductive step (2) above:

*

*What are the maps $\varphi_\alpha$ in the definition? Are they from certain assumptions?

*What are the $\alpha$'s?

*Could one give an example showing what exactly is the quotient space in the definition?

 A: I'll focus on stripping the induction down to its basics (for examples, there's lots in Hatcher).
To start the induction, the discrete topological space $X^0$ is given.
Next, the set of attaching maps $S^0 \to X^0$ is given, one for each 1-cell. Using those attaching maps, the 1-skeleton $X^1$ is constructed as a quotient. Formally an index set is also given, which I'll denote $I_1$, and the attaching maps are denoted $\phi_{\alpha} : S^0 \to X^0$ for each $\alpha \in I_1$.
Next, the set of attaching maps $S^1 \to X^1$ is given, one for each 2-cell. Using those attaching maps, the 2-skeleton $X^2$ is constructed as a quotient. Formally an index set is also given, which I'll denote $I_2$, and the attaching maps are denoted $\phi_{\alpha} : S^1 \to X^1$ for each $\alpha \in I_2$.
Continuing by induction assuming $X^n$ has been constructed.
Next, the next the set of attaching maps $S^n \to X^n$ is given, one for each $n+1$ cell. Using those attaching maps, the $n+1$-skeleton $X^{n+1}$ is constructed as a quotient. Formally an index set is also given, which I'll denote $I_{n+1}$, and the attaching maps are denoted $\phi_{\alpha} : S^n \to X^n$ for each $\alpha \in I_{n+1}$.
and so on ...
