(Kuratowski closure axioms) Let X be a set, and suppose we have a map A ...

 

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(Kuratowski closure axioms) Let X be a set, and suppose we have a map A → A from 2 to 2 satisfying (cf. 2.3.3) i) ACĀ ii) Ā=Ā iii) AUB=AUB iv) = 0. Call U C X open if U = U. True of false: this defines a topology, in which the closure operation is precisely A → A. (Kuratowski closure axioms) Let X be a set, and suppose we have a map A → A from 2 to 2 satisfying (cf. 2.3.3) i) ACĀ ii) Ā=Ā iii) AUB=AUB iv) = 0. Call U C X open if U = U. True of false: this defines a topology, in which the closure operation is precisely A → A.

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False By choosing X 0 and 1 and A 0 and 1 one can create a counterexample In such case Hausdorff topology is not the outcome Let A and X both be 0 and ... View the full answer