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Define a simplicial complex $K$ : If $V=\{v_1,\cdots, v_n\}$ is vertex set and $S$ is a set of some subsets in $V$, then there exists a relation : $$A,\ B\in S \Rightarrow 2^A,\ A\cap B \subset S$$
Define $\underline{K} = (K,|\ \ |)$ : For any $x\in K$, then there exists $A\in S$ s.t. $x\in A=\{ v_{k_1},\cdots,v_{k_i}\}$ and there exist barycentric coordinates for $x$, i.e., $$x=\sum_{j=1}^i \lambda_j v_{k_j},\ \lambda_j\geq 0,\ \sum_{j=1}^i\lambda_j=1$$
Hence we have a metric $$|xy|=\sup_{j} \ \{ |\lambda_j(x)-\lambda_j(y)| \}$$
Then triangulation of $X$ is a homeomorphism $f : \underline{K}\rightarrow X$