Question

topology class

want proof for theorem 7.14 using definition 7.13

please explain well.

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 (T)0-0, A set A C X is first category in X if A-Un=1 An, where each An is nowhere dense in X. If a set is not first category, it is called second category. (Topologically, seoond category sets in X are "thick" and first category sets in X are thin". TO TOPOLOGY Theorem 7.14. X is second category in itself if and only if the intersection of every counfable family of dense open sets in X is nonempty. Definition 7.15. A set A in X is a Gs-set if A-A, where each An is open A set A in X is an Fo"set if A U~1 An, where each An is closed. Theorem 7.16. The complement of a Gs-set is an Fo-set, and vice versa. Theorem 7.17. A closed set in α metric space is α Gé-set. An open set an a metric space is am Fo-set. Theorem 7.18. The irationals in R form a Gs-set Theorem 7.19, A Ga-set in α compact Hausdorff space is a Baire space. Theorem 7.20. The boundary of an open set in X i nowhere dense in X. Theorem 7.21. Every open subset of a Baire space is a Baire space.

Answer 1

Theorem 7:14 port n刁 C0 i S