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5. A topological space (X, ?T ) is said to be a T0-space if for each pair of distinct points a, b in X, either there exists an open set containing a and not b, or there exists an open set containing b and not a. (i) Prove that every T1-space is a T0-space. (ii) Which of (i)–(vi) in Exercise 3 above are T0-spaces? (Justify your answer.) (iii) Put a topology ?T on the set X = {0, 1} so that (X, ?T ) will be a T0-space but not a T1-space. [The topological space you obtain is called the Sierpi?ski space.] (iv) Prove that each of the topological spaces described in Exercises 1.1 #6 is a T0-space. (Observe that in Exercise 3 above we saw that neither is a T1-space.)

          5. A topological space (X, ?T ) is said to be a T0-space if for each pair of distinct points a, b in X, either there exists an open set containing a and not b, or there exists an open set containing b and not a.

(i) Prove that every T1-space is a T0-space.
(ii) Which of (i)–(vi) in Exercise 3 above are T0-spaces? (Justify your answer.)

(iii) Put a topology ?T on the set X = {0, 1} so that (X, ?T ) will be a T0-space but not a T1-space. [The topological space you obtain is called the Sierpi?ski space.]
(iv) Prove that each of the topological spaces described in Exercises 1.1 #6 is a T0-space. (Observe that in Exercise 3 above we saw that neither is a T1-space.)

5. A topological space (X, ?T ) is said to be a T0-space if for each pair of distinct points a, b in X, either there exists an open set containing a and not b, or there exists an open set containing b and not a.

(i) Prove that every T1-space is a T0-space.
(ii) Which of (i)–(vi) in Exercise 3 above are T0-spaces? (Justify your answer.)

(iii) Put a topology ?T on the set X = 0, 1 so that (X, ?T ) will be a T0-space but not a T1-space. [The topological space you obtain is called the Sierpi?ski space.]
(iv) Prove that each of the topological spaces described in Exercises 1.1 #6 is a T0-space. (Observe that in Exercise 3 above we saw that neither is a T1-space.)

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