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Homework 7 Calculus 1
1. Rudin Chapter 2, Problem 9 (a) - (c), and the following statement:
E0( or int(E)) = [
G⊂E, Gopen G.
The problem says that E0 is the largest open subset contained in E. (Compare with Theorem 2.27, which says that the closure ¯E of E is the smallest closed set containing E).
2. Prove that
(a) (0, 1) ⊂ R is not compact. (Note: open 6= not closed.) (b) {n1}∞n=1⊂ R is not compact.
(c) {n1}∞n=1∪ {0} ⊂ R is compact. It is called the compactification of (b).
3. Rudin Chapter 4, Problem 1.
For the following two problems, for f : A → B, define
for E ⊂ A, f (E) := {f (x) | x ∈ E}.
and
for V ⊂ B, f−1(V ) := {x ∈ E | f (x) ∈ V }.
4. Prove that the statement
for all closed subset C ⊂ R, f−1(C) is closed in R is equivalent to
for all open subset E ⊂ R, f−1(E) is open in R
It turns out that both statements are equivalent the fact that f : R → R is continuous (will be shown in class).
5. Rudin, Chapter 4, Problem 3 with X = R. (It might be useful to prove that {a} ⊂ R is a closed set ∀a ∈ R.
6. Given f : [0, 1] → R continuous, and suppose that f ([0, 1]) ⊂ Q. Prove that if f (12) = 0, then f is a constant function.
7. Salas 2.4: 6, 14, 26, 29, 34.
8. Salas 2.6: 5, 9, 10, 15, 25, 26, 28.
