Topology of simplicial complexes

Question

I'm trying to prove that the topology of $|K|$ as a subspace of $\mathbb{R}^{m}$ is the same as the topology of $|K|$ obtained as a quotient space. I was thinking on the equivalence relation of that identification in this way: $ \sigma$ ~ $\tau \iff \sigma \cap \tau \neq \varnothing$, or in the case of abstract simplicial complexes in this way:$\sigma$ ~ $\tau \iff \sigma \subseteq \tau$ or $\tau \subseteq \sigma$. However, those aren't equivalent relations since the transitivity doesn't hold. Can someone please tell me which is the equivalence relation?