Prove that the T_i -property is a topological property for i = 0 Solution Hausdorff spaces do not have in general the homotopy type of any finite space. Recall that a topological space X satisfies the T1- separation axiom if for any two distinct points x, y X there exist open sets U and V such that x U, y V , y / U, x / V . This is equivalent to saying that the points are closed in X. All Hausdorff spaces are T1, but the converse is false.