1. b) Which subsets of R have least upper bounds?
c) Suppose E ⊂ B, lub E = 4, and that x ∉ E if 3 < x < 4. Show that 4 ∈ E.
2. a) For a subset E of a metric space, define
i) interior point of E
ii) E is open
b) Suppose X is a metric space, x ∈ X, y ∈ X, x ≠ y. Show that there are open subsets u, v of X such that x ∈ U, y ∈ V and U ∩ V = ∅.
3. b) Show that [0,2) with the usual metric d(x, y) = |x - y| is not a complete metric space.
4. b) Show that N is not a connected subset of R.
5. b) Using only the definitions in R, show that there cannot be a sequence {Xn} in E with f(Xn) = n². Given that E is compact and f is continuous.
Question
Solution Preview
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.
Return to homework library