By Sloughter D.

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Suppose x = λa + (1 − λ)b for some 0 < λ < 1. 9) so x < b. Similarly, 47 48 CHAPTER 4. TOPOLOGY OF THE REAL LINE so a < x. Hence x ∈ (a, b). Now suppose x ∈ (a, b). Then x= b−x b−a a+ x−a b−a b= and 0< b−x b−a a+ 1− b−x b−a b b−x < 1. D. 1. We say a set U ⊂ R is open if for every x ∈ U there exists > 0 such that (x − , x + ) ⊂ U. 1. Every open interval I is an open set. Proof. Suppose I = (a, b), where a < b are extended real numbers. Given x ∈ I, let be the smaller of x − a and b − x. Suppose y ∈ (x − , x + ).

N}, we say A is finite and write |A| = n. If A has the cardinality of Z+ , we say A is countable and write |A| = ℵ0 . 1. If we define ϕ : Z+ → Z by n−1 , if n is odd, 2 ϕ(n) = − n , if n is even, 2 then ϕ is a one-to-one correspondence. Thus |Z| = ℵ0 . 1) 42 CHAPTER 3. 2. Let A be the set of even integers. Show that |A| = ℵ0 . 3. Verify each of the following: a. If A is a nonempty subset of Z+ , then A is either finite or countable. b. If A is a nonempty subset of a countable set B, then A is either finite or countable.

CARDINALITY and let a1 , a2 , . . , an be the elements of A. Define ϕ : B → P(A) by letting ϕ(b1 , b2 , . . , bn ) = {ai : bi = 1, i = 1, 2, . . , n}. 2) Then ϕ is a one-to-one correspondence. The result now follows from the next exercise. D. 1. With B as in the previous proposition, show that |B| = 2n . In analogy with the case when A is finite, we let 2|A| = |P(A)| for any nonempty set A. 2. Suppose A and B are sets for which there exists a one-toone function ϕ : A → B but there does not exist a one-to-one correspondence ψ : A → B.