I am reading the book "Simplicial objects in algebraic topology" by Peter May and I am trying to understand its proof for Theorem 14.1 on page 56 which says that the geometric realization of a simplicial set is a CW-complex. In this book the proof of this theorem is based on the lemma (Lemma 14.2 on page 56) which says if we denote the geometric realization of the simplicial set $K$ by $\bar{K}$, then any $(k_n,u_n)\in \bar{K}$ is equivalent to a unique non-degenerate point of $\bar{K}$ where we say that a point $(k_n,u_n)\in \bar{K}$ is non -degenerate if $k_n\in K_n$ is non-degenerate and $u_n\in \Delta_n$ is interior. My question is that why this lemma implies that the geometric realization is a CW-complex.
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Did you not read the proof of thm 14.1? What part of the proof did you not understand if you did? – Noel Lundström Jul 10 '22 at 19:34
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There is no proof for that theorem. It is written that the theorem immediately follows from this lemma. – Pouya Layeghi Jul 10 '22 at 19:40
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2Milnor gives a proof in "The Geometric Realization of a Semi-Simplicial Complex," Annals of Math 65 (1957), 357-362. – John Palmieri Jul 11 '22 at 00:07
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Thanks for your comment. I guess Milnor's paper has it as you mentioned. – Pouya Layeghi Jul 11 '22 at 00:31