MATH 214 Final, Fall 2017

Math 214 - Final American University of Beyrouth Fall 2017 - Dr. Richard Aoun Exercise 1. Let X and Y be two topological spaces and f : X −→ Y a map. Show that if X is compact and f is continuous on X, then f(X) is compact (when endowed with the subspace topology). Exercise 2. Parts 1,2 and 3 are mutually independent. We endow R and R2 with the Euclidean topology and their respective subsets with the subspace topology. Recall that S1 denotes the unit circle in R2, i.e. the sphere of center the origin and radius 1 for the Euclidean metric. 1. Let X and Y be two topological spaces. Assume X to be compact and Y to be Hausdorff. (a) Show that any continuous map f : X −→ Y is a closed map. (b) Deduce that any continuous injection from X into Y is an embedding, i.e. yields a homeomorphism between X and f(X). 2. Let f : S1 −→ R be a continuous map. Explain why there exists two real numbers a ≤ b such that f(S1) = [a, b]. 3. Show that there is no homeomorphism between S1 and any closed and bounded interval [a, b]. 4. Deduce from the previous questionS that there is no continuous injection from S1 into R. Exercise 3. Let (X, d) be a metric space. 1. Show that a Cauchy sequence in (X, d) that admits a convergent subsequence is actually convergent. 2. Suppose that there exists some r0 > 0 such that every open ball in (X, d) of radius r0 has compact closure. Show that (X, d) is complete. 3. Suppose that for each x ∈ X, there is an r = r(x) > 0 such that the open ball B(x, r) has compact closure. Show by means of an example that X need not be complete. Check very quickly the properties you are claiming about your spaces. Exercise 4. In this exercise we adopt the following definition of countability: A countable set is a set that is in bijection with a subset of N. Equivalently, it is either a finite set or an infinite one but in bijection with N. You can use without any proof that a countable union (in particular a finite one) of countable sets is still countable. Prove that any compact metric space is separable, i.e. contains a countable dense subset. 1 Exercise 5. (Arzel`a-Ascoli Theorem) Our goal is to prove and discuss a weak form of Arzel`a-Ascoli’s Theorem, which is an important result in functional analysis. Our setting is a compact metric space (X, dX). Denote by C(X) the set of all continuous functions from X to R. We consider the sup-norm d∞(f, g) = sup{|f(x) − g(x)|; x ∈ X}. The aforementioned theorem (see Question 5 for the statement) gives a criteria for a subset of C(X) to be compact. Other than boundness and being closed, an additional assumption should be imposed: that of equicontinuity of a family of continuous functions on X. 1. Explain why d∞ is well-defined and that the “sup” in its definition can be replaced by a “max”. No need to check that it is indeed a metric on C(X). 2. Definition: A subset F ⊆ C(X) is said to be equicontinuous if for every ϵ > 0 there exists δ = δ(ϵ) > 0 such that for every f ∈ F and for every x, y ∈ X, dX(x, y) < δ =⇒ |f(x) − f(y)| < ϵ. (a) Explain why if F is a singleton, i.e. F = {f0} for some continuous function f0 : X −→ R, then F is equicontinuous. In the questions (b), (c) and (d) below, we take X = [0, 1]. (b) Check that, for every C > 0, the set FC of differentiable functions f on [0, 1] such that sup0≤x≤1 |f ′(x)| ≤ C is equicontinuous. (c) Show that F := {x �→ sin(nx); n ∈ N} is a subset of C([0, 1]) which is not equicontinuous. Is the set � C>0 FC of differentiable functions on [0, 1] with bounded derivative equicon- tinuous? Beside a rigorous proof, draw a picture and explain briefly what is happening. (d) Denote by 0 the constant function x �→ 0 on [0, 1]. Show that the closed ball of center 0 and radius 1 in (C([0, 1], d∞) is not equicontinuous. Beside a rigorous proof, draw a picture and explain briefly what is happening. 3. Show that, if a subset F of C(X) is compact, then F is necessarily equicontinuous. 4. ** Let F ⊆ C(X). Show that if F is bounded for the metric d∞ and equicontinuous, then F is totally bounded1. 5. We admit the following criteria of compactness: “a metric space is compact if and only if it is totally bounded and complete”. Deduce from the previous questions Arzel`a-Ascoli’s theorem: F ⊂ C(X) is compact ⇐⇒ F is closed, bounded and equicontinuous. 6. BONUS; BUT WILL BE COUNTED ONLY IF QUESTION 4 IS TREATED. Let n ∈ N∗. Show that Heine-Borel’s theorem in Rn can be deduce from Arzel`a-Ascoli’s theorem, by choosing a suitable compact metric space X. 1Recall that if A is a subset of a metric space (Y, ρ), then A is said to be totally bounded, if for every ϵ > 0 one can find finitely many points a1, · · · , ar ∈ A (that depend on ϵ) such that A ⊆ �r i=1 B(ai, ϵ). 2