PROBLEM 1.

(Diverse questions.) Throughout, (x,d) is a metric space and (E,‖ . ‖E) is a normed vector space (When the context is clear, the subscripts‘x’and‘E’are removed from the distance/norm.)

1. Prove that xis connected if and only if every continuous function x → {0,1} is constant. (Note. {0,1} could be replaced by any discrete set).

2. Here is an attempt to prove that ℝ2 is not homeomorphic to ℝ3 : suppose there is such a homeomorphism 由 ∶ ℝ2 → ℝ3 , and consider a line L in ℝ2 . Then ℝ2 ∖L is not connected whereas ℝ3 ∖由(L) is. Is this reasoning correct? (If not, point out its flaw.)

3. Let (Y,dY) be another metric space. We define the following distance on x × Y d((x,y),(x′ ,y′ )) = max(d (x,x′ ),dY(y,y′ )).

(a) Check that d is a distance on x × Y.

(b) Check that the open sets in x ×Yare exactly the sets of the form U ×V, where U and V are respectively open in x and

.

(c) Let A ⊂ xand B ⊂ Y be connected sets. Prove that A × B is connected in x × Y.

(d)  Same question if we replace connected with compact.

Hint. It is easier to use subsequences...

(e) (Bonus). Prove that a countable product of compact metric spaces is compact.

4.  (a) Suppose that xis connected and let f ∶ x → Y be a continuous function. Prove that if f is locally constant, then it is constant.

Note. f locally constant means thatfor every x0 ∈ x, there exists an open set U containing x0 such that f|U is constant. (b) Determine all the continuous functions f ∶ ℂ∗ → ℂ∗ such that for all z ∈ ℂ∗ , f(z)2 = z2 .

5. Let K,L be compact sets in E.    (a) Prove that K + L is compact.

(b) Define the distance between Kand L as

d(K,L) = inf{‖x − y‖; x ∈ K,y ∈ L}.

Prove that if K and L are disjoint, then d(K,L) > 0.

6.  (Homage to a scribe). Let Mn (ℝ) be the space of n×n, equipped with any norm you like (as you will come to know soon,

they will all be equivalent in this case).

(a) Prove that SLn (ℂ) is path-connected.

(b) Show that SOn (ℝ) = {A ∈ GLn (ℝ); tAA = In , det(A) = 1} is compact.

(c) A matrix A ∈ Mn (K) (K = ℝ or ℂ) is said nilpotent if there exists p ∈ ℕ ∗ such that Ap  = 0. Prove that the set N of nilpotent matrices is path-connected.

Hint. Considerfirst, for any nilpotent matrix A ∈ N, the line segment between A and the zero matrix.

(d) Denote Projn = {P ∈ Mn (ℝ); P2 = P} the set of projection matrices inMn (ℝ). Prove that if two projection matrices P and Q are endpoints of a continuous curve in Projn , then rank(P) = rank(Q).

8. Let Kbe a compact in E, and let p ∶ K → K be function such that ‖f(x) − f(y)‖ < ‖x − y‖, for every x ≠ y ∈ K. Prove that f admits a unique fixed point inK.