This is due to (McCord 67).
For any finite topological space with specialization order , the topological interval map induces a weak homotopy equivalence :
(where we implicitly identify with the category of finite intervals with distinct top and bottom). The isomorphism on the right says that any finite topological space can be constructed by gluing together copies of Sierpinski space, in exactly the same way that any preorder can be constructed by gluing together copies of the preorder .
On the other hand, any finite simplicial complex is homotopy equivalent to its barycentric subdivision, which is the geometric realization of the poset of simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.
A survey is in
The original results by McCord are in
Generalization to ringed finite spaces is discussed in
and aspects of their homotopy theory is discussed in
Any topological meet-semilattice with a bottom element , for which there exists a continuous path connecting to the top element , is in fact contractible. The contracting homotopy is given by the composite . ↩