= Solution
List the finite vertex set as $V=\{v_1,\ldots,v_N\}$. Map a point to its <barycentric coordinates>:
$$
\boxed{f:|K|\longrightarrow\mathbb R^N,\qquad f\left(\sum_jt_jv_j\right)=(t_1,\ldots,t_N).}
$$
On each simplex this is an <affine map> and therefore <continuous>. The definitions agree on every common face, so the <quotient topology> defining the realization makes $f$ continuous on all of $|K|$. Barycentric coordinates are unique after zero coordinates are added, so $f$ is <injective>.
Since <Euclidean space> is <Hausdorff>, distinct images have disjoint open neighbourhoods. Their inverse images separate the original points. Hence \b[$|K|$ is <Hausdorff>]. A continuous injection into a Hausdorff space is enough for this conclusion; no prior Hausdorff assumption on the domain is being used.
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