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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

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Prove that the T_i -property is a topological property for i = 0S.pdf

  • 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 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