understanding the definition of CW complexes

Question

I am trying to read Bredon's Topology and Geometry and this is what I understand of the definition of a CW complex.

Let $K^{0}$ be a discrete set of points, i.e. points under the discrete topology. These points are called 0-cells.

We shall construct $K^{n}$ inductively.

Before that, we shall define what it means to write $Y \cup_{f} X$. Suppose, $A \subset X$ is a subspace and $f:A \to Y$. We identify $a \sim f(a)$ for $a \in A$ and regard the rest of the singleton points in the disjoint union $X \sqcup Y$ as distinct equivalence classes (Note that $\sim$ defines an equivalence relation on $X \sqcup Y$). Then we define $Y \cup_{f} X$ to be the quotient space $X \sqcup Y / \sim$.

Now, back to our construction.

Suppose, $K^{n-1}$ has been defined. Let $\{f_{\partial \sigma}\}$ be a collection of maps $f_{\partial \sigma}:S^{n-1} \to K^{n-1}$ where $\sigma$ ranges over some indexing set. Let $\displaystyle Y=\sqcup_{\sigma} D_{\sigma}^n$ (disjoint union) where each $D_{\sigma}^n$ is a copy of the $n-$disc, $D^n$.

We put together a map $f:B \to K^{n-1}$ by taking $x \in D_{\sigma}^n$ to $f_{\partial \sigma}(x)$, where $B=\sqcup S_{\sigma}^{n-1}$ (am I right?).

Then we define $$K^n=K^{n-1}\cup_{f}Y.$$

Once we have done this, $K:=\cup_{n}K^{n}$. We define the topology on $K$ as follows: $A \subset K$ is said to be open iff $A\cap K^{n}$ is open in $K^{n}$.

Next, Bredon says something like :

For each $\sigma$ let $f_{\sigma}:D_{\sigma}^n \to K$ be the canonical map.

My questions are the following:

1. Is my construction of $f$ correct?
2. What is this canonical map exactly? I can see that if there is such a canonical map from the domain to some $K^n$, we can get a map to $K$ by inclusion, but I don't see where the points in $D^n_{\sigma}-S^{n-1}_{\sigma}$ should go to.
3. Usually,when the author says a map is canonical, one reason why readers don't get what it means is when they don't understand the intuition behind something. I am guessing I am missing that here?

Thank you so much in advance. It's probably not a very bright question.

Answer 1

You forgot to say want $B$ is, I assume that it is the subset of $Y$ corresponding to the boundaries of the discs. You want to use $f_{\partial\sigma}$ in the definition of $f$ then.

For the definition of $f_\sigma$, just map to the part of $Y$ indexed by $\sigma$ and then to the quotient $K^{n}$ then to $K$.