In spanier, there are two different topologies on a simplicial complex $K$. (I don't know if this is also the case in other books, Spanier is the only book I am reading.)
First topology is a metric topology with distance defined by $d(\alpha,\beta) = \sqrt{\sum_{v \in K} [\alpha(v) - \beta(v)]^2}$, denoted in the book as $|K|_d$. The second topology is the topology coherent with $\{|s|_d; s \in K\}$.
Since the book provides no theorem proving the equivalence of the two topologies, I assume they are different. Intuitively, things should go wrong when $K$ has infinite dimension, but I have trouble constructing one. Any ideas or suggestions?