9.D. What are the interior, boundary, and exterior points in R of the set [0, 1)....

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9.D. What are the interior, boundary, and exterior points in R of the set [0, 1). Conclude that it is neither open nor closed. 9.E. Give an example in R² which is neither open nor closed. Prove your assertion. 9.F. Write out the details of the proof of Theorem 9.6. 9.G. Show that a subset of Rº is open if and only if it is the union of a countable collection of open balls. (Hint: the set of all points in Rº all of whose coordinates are rational numbers is countable.) 9.H. Every open subset of Rº is the union of a countable collection of closed sets. 9.1. Every closed subset of R is the intersection of a countable collection of open sets. 9.J. If A is any subset of Rº, let Aº denote the union of all open sets which are contained in A; the set Aº is called the interior of A. Note that A° is an open set; prove that it is the largest open set contained in A. Prove that Aº≤ A, (A°)⁰ = Aº (AnB)=A°n B, (RP)⁰ = RP. Give an example to show that (AUB) = A°UB° may not hold. 9.D. What are the interior, boundary, and exterior points in R of the set [0, 1). Conclude that it is neither open nor closed. 9.E. Give an example in R² which is neither open nor closed. Prove your assertion. 9.F. Write out the details of the proof of Theorem 9.6. 9.G. Show that a subset of Rº is open if and only if it is the union of a countable collection of open balls. (Hint: the set of all points in Rº all of whose coordinates are rational numbers is countable.) 9.H. Every open subset of Rº is the union of a countable collection of closed sets. 9.1. Every closed subset of R is the intersection of a countable collection of open sets. 9.J. If A is any subset of Rº, let Aº denote the union of all open sets which are contained in A; the set Aº is called the interior of A. Note that A° is an open set; prove that it is the largest open set contained in A. Prove that Aº≤ A, (A°)⁰ = Aº (AnB)=A°n B, (RP)⁰ = RP. Give an example to show that (AUB) = A°UB° may not hold.

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A The closed set 01 is a subset of R and it is closed because it contains all of its limit points It is not open because it does not contain any of its interior points B A set U in R is open if for ev... View the full answer