How does one give topological structure to an abstract simplicial complex?

Question

Given an abstract simplicial complex $K,$ I'd like to know how I can endow it with a topology.

Answer 1

It may be convenient to endow the vertex set $V\cal(A)$ with a partial order, so that the vertices of each simplex are totally ordered. Now for each $n$-simplex $σ=[v_0...v_n]$ take a copy of the standard $n$-simplex $\Delta^n $. We will denote the vertices of $Δ^n$ by $v_i$, too, according to their ordering, so $v_i=(0,...,1,...0)$ will be the $i$-th standard basis vector of $\Bbb R^{n+1}$. The subsimplices of $\sigma$ correspond to order-preserving injections into $[n]$, that means $[w_0...w_k]=[v_{f(0)}...v_{f(k)}]\triangleq f:[k]\to[n]$. Now identify $Δ^k$ with $Δ^n$ via the linear map sending $w_i$ to $v_{f(i)}$. The topological space $|\cal A|$ is the quotient of the disjoint union of all these simplices under the relation generated by these embeddings.