Introduction to Analysis
Homework 2
1. (Rudin ex2.8) Is every point of every open set E ⊂ R2 a limit point of E? Answer the same question for closed sets in R2.
2. (Rudin ex2.9) Let E◦ denote the set of all interior points of a set E. [E◦ is called the interior of E.]
(a) Prove thatE◦ is always open.
(b) Prove the E is open if and only if E◦ =E.
(c) IfG⊂E and G is open, prove that G⊂E◦.
(d) Prove that the complement of E◦ is the closure of the complement ofE.
(e) Do E and ¯E always have the same interiors?
(f) Do E and E◦ always have the same closures?
3. (Rudin ex2.10) LetX be an infinite set. For p∈X and q ∈X, define
d(p, q) =
1 (ifp6=q) 0 (ifp=q).
Which subsets of the resulting metric space are open? Which are closed? Which are compact?
4. GivenR1 with the metricd(x, y) =|x−y|. Show that a finite set consists only of isolated points. Is it true that a set consisting only of isolated points must be finite?
5. Construct a set which consists only of isolated points but is not closed.
6. (Rudin ex2.13) Construct a compact set of real numbers which limit points form a count- able set.
7. Prove that the union of finite compact sets is compact. Is the union of infinite compact sets compact?
8. Let A be the space of sequences x= {x1, x2,· · ·, xn,· · · } in which only a finite number of the xi are different from zero. In A, define d(x,y) by the formula
d(x,y) = max
1≤i<∞|xi−yi|.
(a) Show thatA is a metric space.
(b) Find a closed bounded set inA which is not compact.
