A set Y X is called an open domain if Y = Int(Y ). A set Y is called a closed domain if Y = Int(Y ). Prove the following. (30 points)

(i) If Y is a closed set, then Int(Y ) is an open domain.

(ii) If Y1, Y2 are open domains, then Y1 Y2 is an open domain.

(iii) If Y1, Y2 are open domains, is it true that Y1 Y2 is an open domain? If yes, prove it, if no, provide one example to show it is not true.

(iv) Y is a closed domain, if and only if Y c = X Y is an open domain.

(v) Consider R with the standard topology. Is Q an open domain? Is Q a closed domain?

Justify your answer.

2. QUESTION 2 A set Y C X is called an open domain if Y = Int(Y). A set Y is called a closed domain if Y = Int(Y). Prove the following. (30 points) (i) If Y is a closed set, then Int(Y) is an open domain. (ii) If Y1, Yz are open domains, then YAY, is an open domain. (iii) If Yi, Yz are open domains, is it true that YUYz is an open domain? If yes, prove it, if no, provide one example to show it is not true. (iv) Y is a closed domain, if and only if Yo = X - Y is an open domain. (v) Consider R with the standard topology. Is Q an open domain? Is Q a closed domain? Justify your