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?
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 ...