at.algebraic topology - Why is complex projective space triangulable?

I will present a triangulation of $mathbb{CP}^{n-1}$. More specifically, I will give an explicit regular CW structure on $mathbb{CP}^{n-1}$. As spinorbundle says, the first barycentric subdivision of a regular CW complex is a simplicial complex homeomorphic to the original CW complex.

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Recall that to put a regular CW complex on space $X$ means to decompose $X$ into disjoint pieces $Y_i$ such that:

(1) The closure of each $Y_i$ is a union of $Y$'s.

(2) For each $i$, the pair $(overline{Y_i}. Y_i)$ is homemorphic to $(mbox{closed} d-mbox{ball}, mbox{interior of that} d-mbox{ball})$ for some $d$.

The barycentric subdivision of $X$ corresponding to this regular CW complex is the simplicial complex which has a vertex for each $Y_i$ and has a simplex $(i_0, i_1, ldots, i_r)$ if and only if $overline{Y_{i_0}} subset overline{Y_{i_1}} subset cdots subset overline{Y_{i_r}}$.

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Write $(t_1: t_2: ldots: t_n)$ for the homogeneous coordinates on $mathbb{CP}^{n-1}$. For $I$ a nonempty subset of ${ 1,2, ldots, n }$, let $Z_I$ be the subset of $mathbb{CP}^{n-1}$ where $|t_i|=|t_{i'}|$ for $i$ and $i' in I$ and $|t_i| > |t_j|$ for $i in I$ and $j not in I$. Note that $Z_I cong (S^1)^{|I|-1} times D^{2(n-|I|)}$, where $D^k$ is the open $k$-disc. Also, $overline{Z_I} = bigcup_{J supseteq I} Z_J cong (S^1)^{|I|-1} times overline{D}^{2(n-|I|)}$ where $overline{D}^k$ is the closed $k$-disc.

We now cut those torii into discs. For $i$ and $i'$ in $I$, cut $Z_I$ along $t_i=t_{i'}$ and $t_i = - t_{i'}$. So the combinatorial data indexing a face of this subdivision is a cyclic arrangement of the symbols $i$ and $-i$, for $i in I$, with $i$ and $-i$ antipodal to each other. For example, let $I={ 1,2,3,4,5 }$ and write $t_k=e^{i theta_k}$ for $k in I$. Then one of our faces corresponds to the situation that, cyclically,
$$theta_1 < theta_2 = theta_4 + pi < theta_3 = theta_5 < theta_1+ pi < theta_2 + pi = theta_4 < theta_3 + pi = theta_5 + pi < theta_1.$$
This cell is clearly homeomorphic to ${ (alpha, beta) : 0 < alpha < beta < pi }$. Similarly, each of these cells is an open ball, and each of their closures is a closed ball. We have put a CW structure on the torus.

Cross this subdivision of the torus with the open disc $D^{2(n-|I|)}$. The result, if I am not confused, is a regular $CW$ decomposition of $mathbb{CP}^{n-1}$.