Question

topology class

want proof for theorem 7.16 using definition 7.15

Definition 7.13. X is a Baire space if the intersection of each countable family of dense open sets is dense. A set A C X is nowhere dense in X if (A)A set ACXis first category in X if AAn, whcre cach An is nowbere dense in X. If a set is not first category, it is called second category. (Topologically, second category sets in X are thick" and first category sets in X are "thin") Theorem 7.14. X is second category in itself if and only if the intersection of every countable family of dense open sets in X is noempty Definition 7.15. A set. A in X is a Ga-set if A-ng.' A,, , where each A갸 is open. A set A in X is an Fey-set if A =嶩1 An, where each A" is ched. Theorem 7.16. Thecomplement of a Gs-set is an Fo set, and vice versa. Theorem 7.17. A closed set in a metric space is a Gsse. An open set in a metric spoor is an Fe-set. Theorem 7.18. The irationals in R form a Gs-set Theorem 7.19. A Gs-set in a compact Hausdor space is a Baire space Theorem 7.20. The boundary of an open set in X is nowhere dense in X Theorem 7.21. Beery open subset of a Baire space is a Baire space.

Answer 1

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