Years ago, my advisor showed me a construction where you take a CW complex and quotient each open cell to a single point. He said that under certain conditions (I believe always satisfied by the second barycentric subdivision of a complex) the resulting space would be a finite non-Hausdorff space homotopy equivalent to the original space.

**Edit** By 'homotopy equivalent' I really mean they have identical homotopy groups. I don't think this is equivalent to regular homotopy equivalence in the non-Hausdorff case.

He said this was well-known although not very popular. What is the name for this construction? What is a reference for the homotopy equivalence result?