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 Xsatisfies the T1- separation axiom if for any two distinct points x, y X there exist open setsu and V such that x U,y V , y / U, x / V . This is equivalent to saying that the points are closed in X. AllHausdorff spaces are T1,but the converse is false..

1. 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
Xsatisfies the T1- separation axiom if for any two distinct points x, y X there exist open setsu
and V such that
x U,y V , y / U, x / V . This is equivalent to saying that the points are closed in X. AllHausdorff
spaces are
T1,but the converse is false.