Description

• Write in a neat and legible handwriting or use LATEX
• Clearly explain the reasoning process
• Write in a complete style (subject, verb, and object)
• Be critical on your results
Questions preceded by a * are optional. Although they can be skipped without any deduction, it is important to know and understand the results they contain.
Ex. 1 — Connected space
In this exercise we provide alternative definitions for a connected space. Let X be a metric space. We want to prove that the following conditions are equivalent
(i) The only subsets that are both open and closed in X are X and ∅;
(ii) It is impossible to write X as the union of two disjoint, non-empty open subsets;
(iii) It is impossible to write X as the union of two disjoint non-empty closed subsets;
(iv) There is no continuous, surjective application from X into {0,1}⊂R;
1. Prove that (i), (ii), and (iii) are equivalent.
* 2. Assume (iv) is false and show that (iii) is false.
* 3. Assume (iii) is false and show that (iv) is false.
Ex. 2 — Intermediate value theorem
In this exercise we prove the intermediate value theorem following the sketch of proof provided in the slides (proof 2.11). From a high level perspective the result relies on the connectedness property of the real numbers.
1. Let X and Y be two metric spaces, A be a connected subset of X, and f : X → Y be a continuous map. Show that f (A) is connected.
Hint: use one of the characterisations of a connected set from exercise 1.
2. Let A be a subset of R. We want to prove that A is connected if and only if A is an interval.
a) Show that it is true for the empty set, and for all the subsets of R composed of a single element.
b) Assuming that A is not an interval prove that A is not connected.
c) We now prove the converse.
i – Let I and J be two non-empty open intervals of R. Show that there exists a continuous bijection from I into J whose inverse is also a continuous bijection.
ii – Using question 1, show that it suffices to prove that R is connected.
* iii – Let U be a subset of R, different from R, that is both open and closed in R. Find a contradiction and conclude that R is connected.
Hint: observe that a closed and non-empty set having an infimum has a minimum.
3. Conclude the proof of the intermediate value theorem in the case of the real numbers.
Ex. 3 — Rolle’s theorem
Reasoning by induction and applying the extreme values theorem, prove Role’s theorem (theorem 2.18).
Ex. 4 — Extreme value theorem
In this exercise we prove the extreme value theorem following the sketch of proof provided in the lecture slides (proof 2.14).
1. Let X be a metric space and A a subset of X.
a) Show that if A is a compact subset of X then A is closed in X.
b) Prove that if a subset of R is compact then it is closed and bounded.
2. Keeping the same notations we now prove the converse.
a) Prove that if X is compact and A is closed in X, then A is a compact subset of X.
* b) We want to prove that for any a ≤ b ∈R, [a,b] is compact in R. Let L = [a,b] et (Ui)i∈I be a family of opens from R covering L. We define A as the set of x ∈ L such that [a,x] is covered by a finite number of Ui. i – Prove the result for a = b.
ii – We now treat the case where a ∈ A and a < b. Let m be the supremum of A. Show that m ∈ A.
iii – Assume m < b, and show the existence of y ∈ A such that y > m.
iv – Conclude that if A is closed and bounded then it is compact.
3. Complete the proof of the extreme value theorem.
Ex. 5 — Continuity
In this exercise we provide alternative characterisations for a continuous function and in particular complete proof 2.10. Let X and Y be two metric spaces, f be a function from X into Y , and a ∈ X. We want to prove that the following conditions are equivalent
(i) For all , there exists such that f (B(a,η)) ⊂ B(f (a),ε);
(ii) For all , there exists such that d(f (x),f (a)) <ε when d(a,x) <η, for x ∈ X;
(iii) For any neighborhood V of f (a), there exists a neighborhood U of a such that f (U) ⊂ V ;
(iv) For any neighborhood V of f (a), f −1(V) is a neighborhood of a;
1. Show that (i) and (ii) are equivalent;
* 2. Consider a neighborhood of f (a) and prove that (i) implies (iii).
* 3. Observe that if V is a neighborhood of a ∈ X, then any subset of X containing V is a neighborhood of a. Conclude that (iii) implies (iv).
4. Consider V = B(f (a),ε), for some ε, and prove that (iv) implies (i).