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In class, I was given a constructive definition of a CW-complex and told that it's an easy exercise to prove that it is Hausdorff. I included the definition given in class below;

We set $X^0 = \bigsqcup_{i\in I_0}D^0_i$. That is, $X^0$ is the union of disjoint 0-balls (i.e. points). Now, for each $n \in \mathbb{N}$, if $X^{n-1}$ is given then we define attaching maps for each $i$ in some index set $I_n$; $$ f_i^n:\partial D_i^n \to X^{n-1} $$ Then $$ X^n = \left.\left(X^{n-1}\sqcup \bigsqcup_{i\in I_n}B_i^n\right)\middle/ \left\{p\sim f_i^n(p) : p\in \partial D_i^n, i\in I_n\right\}\right. $$

Being very new to algebraic topology, I do not see how the proof is obvious. The statement seems clear, but I am having a hard time using the definition to construct a rigorous proof. If anyone could help me prove this (and maybe elaborate on where detail is needed) I would appreciate it.

Thanks in advance!

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  • $\begingroup$ Check Proposition A.3 page 522 in Hatcher's Algebraic Topology. If there's something you can't untangle in the proof, come back for help. Here's a link to Hatcher's book, with no CopyRight infringement pi.math.cornell.edu/~hatcher/AT/AT.pdf $\endgroup$
    – Laz
    Oct 1, 2018 at 22:36
  • $\begingroup$ @Laz I don't understand what he means by $(1-\epsilon_\alpha, 1]\times \Phi^{-1}_\alpha(N_\epsilon^n(A))$. Could you elaborate on that part? Also, Hatcher defines a CW-complex as the union of all the $X^n$s, the definition we have does not mention that. I'm still unclear about whether I should view $X^{n-1}$ as a subset of $X^n$ or if they are completely different since the latter is a quotient space. $\endgroup$
    – Quoka
    Oct 1, 2018 at 22:43
  • $\begingroup$ I'm sorry I don't have the time for a full answer now, but here's some. If $A$ is a closed of $X$ (you may take both $A$, $B$ in the proof to be points, since you want only Hausdorff and not "normalness"), the $N_{\epsilon}^n(A)$'s are open neighborhhods of $A\cap X^n$ in $X^n$, where $X^n$ is the $n$-skeleton, or the union of all cells of dimensions not greater than $n$, constructed inductively.The $\Psi_\alpha$ is just the adjunction map of the cell $e_{\alpha}^n$ followed by the injection of the $n$-skeleton $X^n$ in $X$. Check how is defined the topology of $X$ using that of the $X^n$'s. $\endgroup$
    – Laz
    Oct 1, 2018 at 23:00
  • $\begingroup$ Start reading from page 519, so that you get a good feeling of this, and understand the difference between the two ways to define a CW complex. These are delicate matters, so invest some time on it once and for all. If necessary, in Chapter $0$ there's a little intro to it. $\endgroup$
    – Laz
    Oct 1, 2018 at 23:03
  • $\begingroup$ Strangely, I find it a bit easier to prove that a CW-complex is normal by using the definition to construct maps $X\to [0,1]$ separating two closed sets. $\endgroup$ Oct 2, 2018 at 9:46

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