Finite simplicial weak topology agrees with euclidean topology

In a finite nonempty vertex set, each simplex is defined by nonnegative coordinates, sum $1$, and zero coordinates outside its face. It is closed and bounded in the ambient finite-dimensional Euclidean space, hence compact. The finite union $\mathrm{\mid }K\mathrm{\mid }$ is also closed and bounded, hence compact by Heine–Borel. If $K$ has no vertices its realization is empty and compact directly.

If $E\subseteq \mathrm{\mid }K\mathrm{\mid }$ is weakly closed, then $E\cap \mathrm{\mid }\sigma \mathrm{\mid }$ is closed in the Euclidean simplex, and thus closed in the ambient Euclidean space since $\mathrm{\mid }\sigma \mathrm{\mid }$ is closed. Their finite union is $E$, so $E$ is Euclidean closed. Conversely, a Euclidean relatively closed $E$ has closed traces on every simplex and is weakly closed. Consequently both topologies agree, and the Euclidean metric and Hausdorff property restrict to $\mathrm{\mid }K\mathrm{\mid }$.

For finite $P\subseteq K$ and closed $E\subseteq \mathrm{\mid }P\mathrm{\mid }$, write $E={\bigcup }_{\sigma \in P}(E\cap \mathrm{\mid }\sigma \mathrm{\mid })$. For any $\tau \in K$, each summand meets $\mathrm{\mid }\tau \mathrm{\mid }$ in a closed subset of the common face $\mathrm{\mid }\sigma \cap \tau \mathrm{\mid }$, hence in a closed subset of $\mathrm{\mid }\tau \mathrm{\mid }$. There are finitely many summands, so $E$ is weakly closed in $\mathrm{\mid }K\mathrm{\mid }$. Conversely an ambient weakly closed set has closed traces on the simplices of $P$. These two implications show that the inclusion induces exactly the topology of $\mathrm{\mid }P\mathrm{\mid }$ and is closed; taking $E=\mathrm{\mid }P\mathrm{\mid }$ proves the closed-image assertion.