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As a beginner, I am struggling a bit with CW-complexes. I'm reading Hatcher, chapter 0. So I want to pose a few questions that are almost embarrassing to me but I believe it is important to ask such questions.

1) As is usual, Hatcher defines the $n$-skeleton as $\left(X^{n-1} \bigsqcup_{\alpha} D_{\alpha}^n \right)/\sim$ with the usual equivalence relation. He later says that as a set we have $$X^n = X^{n-1} \bigsqcup_{\alpha} e_{\alpha}^n $$ where $e_\alpha^n$ is an open $n$-disc.

Does he mean that points on the boundaries of $D_{\alpha}^n$ are identified with points in $X^{n-1}$ and so we can think of these points as points on $X^{n-1}$? So what remains of $D_{\alpha}^n$ are the interiors, i.e. $e_{\alpha}^n$...

(This agrees with the pictorial intuition of gluing.)

2) Why do we define the characteristic map $\Phi_{\alpha} : D_{\alpha}^n \to X$? Where is this used? Maybe is better to first ask the more naive question: What does this map even do intuitively/geometrically?

3) A subcomplex $A$ is a closed subspace of a CW-complex $X$ that is an union of cells in $X$. Hatcher says that the image of characteristic functions of cells in $A$ are again in $A$ because $A$ is closed.

Why do we need closedness here?

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  • $\begingroup$ Please restrict questions to one per post. $\endgroup$ – amWhy Feb 26 at 12:22
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    $\begingroup$ How well do you understand quotient topologies? I ask because your questions seem to skip over every point in Hatcher's text where quotient topologies are relevant. For example, you don't mention the attaching maps $\phi^n_\alpha$, nor how they are used to define the identifications $\sim$ when forming the quotient topology on $\left(X^{n-1} \bigsqcup_{\alpha} D_{\alpha}^n \right)/\sim$, nor the relation between the attaching maps $\phi_\alpha = \phi^n_\alpha$ and the characteristic maps $\Phi_\alpha$. $\endgroup$ – Lee Mosher Feb 26 at 12:24
  • $\begingroup$ @amWhy I would argue that these questions are related enough to be posed in the same thread. $\endgroup$ – Qi Zhu Feb 26 at 12:27
  • $\begingroup$ @LeeMosher I would hope well enough but I'm sure I can always improve. I made the decision not to mention those things because I thought it might would have been counterproductive to just retell everything about these definitions. $\endgroup$ – Qi Zhu Feb 26 at 12:30
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Regarding your question 1), Hatcher says, on page 5 item (2):

... form the $n$-skeleton $X^n$ from $X^{n-1}$ by attaching $n$-cells $e^n_\alpha$ via maps $\phi_\alpha : S^{n-1} \to X^{n-1}$. This means that $X^n$ is the quotient space of the disjoint union $X^{n-1} \bigsqcup_{\alpha} D_{\alpha}^n$ of $X^{n-1}$ with a collection of $n$-discs $D^n_\alpha$ under the identifications $x \sim \phi_\alpha(x)$ for $x \in \partial D^n_\alpha$.

So, as you say, it really is just an issue of retelling the definition: points on the boundaries of $D^n_\alpha$, namely the points $x \in \partial D^n_\alpha = S^{n-1}$, are identified with points in $X^{n-1}$, namely $x$ is identified with $\phi^n_\alpha(x) \in X^{n-1}$. One could also write $\partial D^n_\alpha = S^{n-1}_\alpha$ to resolve a little bit of ambiguity.

Regarding your question 2), you will find, as you read on, that the characteristic maps are used in various important places. They are of chief importance in discussions of homology later on. If you have an on-line version of the book and search for "characteristic map", you can find some of those places, for example: starting on page 105 at the beginning of the discussion of simplicial homology; starting on page 128 when proving the equivalence of simplicial homology and singular homology; on page 141 when proving equivalence of CW and singular homology.

From an intuitive point of view, the characteristic maps tell you how the open cells are related to the rest of the complex. An open $n$-cell $e^n_\alpha$ is not just floating somewhere disconnected from the $n-1$ skeleton. Instead, one can say intuitively that $e^n_\alpha$ is bonded to the $n-1$ skeleton by the characteristic map (see Hatcher's discussion of characteristic maps on page 7). One say also say rigorously that the closure in $X$ of the open cell $e^n_\alpha$ is equal to $\Phi^n_\alpha(D^n_\alpha) = e^n_\alpha \cup \phi^n_\alpha(S^{n-1}_\alpha)$, which is a consequence of the fact that $\Phi^n_\alpha : D^n_\alpha \to X$ is continuous and that the closure in $D^n_\alpha$ of its interior $\text{int}(D^n_\alpha)$ is equal to $D^n_\alpha = \text{int}D^n_\alpha \cup S^{n-1}_\alpha$.

Regarding your question 3), since $\Phi^n_\alpha(D^n_\alpha)$ is the closure of $e^n_\alpha$, it follows that if a closed subset $A \subset X$ contains $e^n_\alpha$ then $A$ contains $\Phi^n_\alpha(D^n_\alpha)$.

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  • $\begingroup$ Thank you, Prof. Mosher. That helped a lot! I like that you pointed out that I could just search for "characteristic map" in the electronic version (and in a printed book also in the index) - I wonder why I forgot to do that. But in particular, I think you explained very well that the point of characteristic maps is to say how the open cells are related to the rest of the complex. $\endgroup$ – Qi Zhu Feb 27 at 20:06

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