Simplicial topology

Let K be a simplicial complex, and Δ ∈ K a simplex. Then the topology on Δ as a subspace topology of |K| is the same as the topology on Δ as a subspace topology of the Euclidean space in which K is embedded. So a subset of Δ is |K|-closed if and only if it is Euclidean-closed, and in particular, Δ is a closed subset of |K|.