P is regular if all Borel sets are regular. Define C to be the collection of regular sets. (a) Show R e C, Ø e C. (b) Show C is closed under complements and countable unions. (c) Let F(R) be the closed subsets of Rk. Show

Prove directly that the union of two disjoint sets, where one is countably infinite and the other is finite, is countable.

A set F is called an Fo-set (or an Fo) if it is a countable union of closed sets. Likewise, a set G is called a Gs-set (or a Gs) if it is a countable intersection of open sets. Then show that (a) [0, 1] is a Gs set. (b) (a,b] is a both Gs and an Fo set. (c) Q is an F set, but not a Gs set.

Let X and Y be sets. Let us say that a subset S of the cartesian product X x Y is a rectangle if S = A × B for some A C X and BCY. 10. (a) Prove that every one-element subset of X x Y is a rectangle. (b) Prove that whenever X is a set with more than one element, the subset {(x, æ) : ¤ € X} S = of X x X is not a rectangle. (Here Y = X.) (c) Is the union of two rectangles in X x Y always a rectangle? Give either a proof that this is true or an example to show that it is false.