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So, for a while now I've been looking for a short but concise proof of the fact that

Every CW complex is paracompact.

I finally found the following proof (shortest so far) of this theorem but couldn't understand a few things the author did here:

enter image description here enter image description here

The lemma being referred to (4th line from below) is as follows:

Lemma. Let $K$ be compact, $C\subseteq K$ be closed, and $\mathcal U=\{U_\alpha\}$ be an open cover of $K$. If $\{\psi_\alpha\}$ is a partition of unity subordinate to $\{C\cap U_\alpha\,:\,U_\alpha \in\mathcal U\}$, then there exists a partition of unity $\{\Psi_\alpha\}$ subordinate to $\mathcal U$ s.t. $\Psi_\alpha\big|_C=\psi_\alpha$ for each $\alpha$.

Also, the carrier of $x$, written $C(x)$, is defined as follows:

Definition. If $(X,\mathcal E)$ is a cellcomplex, the carrier of $A\subseteq X$, $C(A)$, is the intersection of all subcomplexes of $(X,\mathcal E)$ containing $A$.

I have understood all the individual logical steps in the proof. The part that confuses me is the following:

The author proved in the end that $(X_{\Gamma_0},U_{\Gamma_0},p_0)$ cannot be maximal in $T$. How does this imply that there is a partition of unity subordinate to $\mathcal U$?

Maybe I'm just overlooking something very trivial here! The proof is taken from Lundell's The Topology of CW Complexes.

P.S.: I'm sorry for the image but typing the entire proof would have been too tedious.

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1 Answer 1

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The final step of the proof showed that $(X_{\Gamma_0},U_{\Gamma_0},p_0)$ cannot be maximal if $X_{\Gamma_0}\neq X$. So, the maximal element $(X_{\Gamma_0},U_{\Gamma_0},p_0)$ given by Zorn's lemma must satisfy $X_{\Gamma_0}=X$. This means exactly that $p_0$ is a partition of unity subordinate to $\mathscr{U}$ on all of $X$.

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  • $\begingroup$ Of course! I completely overlooked that. Thanks a lot! $\endgroup$ Jul 2, 2023 at 1:42

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