Consider a simplicial complex $C$. On its support $$|C|=\lbrace \alpha = \sum_{v\in C}\alpha_{v}v \mid 0\leq \alpha_{v} \leq 1 , \sum_{v\in C}\alpha_{v} =1\mbox{ and }v|{\alpha_{v}} \neq 0\mbox{ is a simplex in }C \rbrace $$ there is a metric, the barycentric metric, defined by $d(\alpha,\beta)=\sqrt{\sum_{v\in C}(\alpha_{v} - \beta_{v})^{2}}$. Is this metric complete ?

Greetings Ben