I have a fundamental problem in the proof of Theorem 5.20 (J.M.Lee: "Introduction to Topological Manifolds" - pag.138/139).
The statement:
$Suppose \quad\text{} X_0\subseteq X_1\subseteq ...\subseteq X_{n-1}\subseteq X_{n}\subseteq....$ is a sequence of topological spaces satisfying the conditions:
(i) $X_{0}$ is a nonempty discrete space.
(ii) For each n$\geq1$, $X_n$ is obtained from $X_{n-1}$ by attaching a (possibly empty) collections of n-cells.
$Then\quad\text{} X = \bigcup_{n}X_n\quad\text{}$ has a unique topology coherent with the family$\quad\text{}$ {$X_n$} , and a unique cell decomposition making it into a CW complex whose n-skeleton is $X_n$ for each n. (end of the statement)
My only problem at the beginning of the proof of this theorem is that prof. Lee quickly concludes that $X_{n-1}$ is closed in $X_n$ for each n.
I reason as follows: there is quotient map - say $q_{n}$:
$q_n$ = $\quad\text{}X_{n-1}\bigsqcup (\bigsqcup\limits_\alpha (D_\alpha)\longrightarrow X_{n-1}\cup_\varphi(\bigsqcup\limits_{\alpha} D_\alpha)$
in which: {${D_\alpha}$} is a collection of closed n-cells (indexed by $\alpha$) and $\varphi = \bigsqcup_\alpha \delta D _ \alpha\longrightarrow X_{n-1}$ $\quad\text{}$the appropriate attaching function.
The definition of $\quad$ "$X_n$ is obtained from $X_{n-1}$ by attaching ..."$\quad$ in (ii) as given by prof. Lee (end of pag.129) says: $X_n$ is homeomorphic to the adjunction space $X_{n-1}\cup_\varphi(\bigsqcup\limits_{\alpha} D_\alpha)$ hereabove.This means that there is a function -say $\phi$- which connects the adjunction space and $X_n$ homeomorphically:
$\phi$ = $\quad\text{}X_{n-1}\cup_\varphi(\bigsqcup\limits_{\alpha} D_\alpha)\longrightarrow$ $X_n$
(Prof. Lee does not write this $\phi$ explicitly).
Now, from Prop. 3.77 (pag.73) we know that$\quad\text{}q_n(X_{n-1})$ is closed in$\quad\text{}$ $X_{n-1}\cup_\varphi(\bigsqcup\limits_{\alpha} D_\alpha)$ and thus $\quad\text{}(\phi o q_n)(X_{n-1})$ is closed in $X_n$. This however does not coincide with the desired result that $X_{n-1}$ is closed in $X_n$.
It seems to me that the composite$\quad\text{}(\varphi o q_n)$ is somehow considered as an identity function or that the subsetsequence in the statement is not literally considered as subsets (?).
Comments are greatly appreciated.
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