HOMEWORK 2 — DUE OCT 3RD

MATH 25

There are three sections: A,B and C (plus a fourth “optional” one which won’t be graded). Please return the three parts separately to the proper CA.

A. Problems graded by Jennifer

A.1. Give an example of a metric space (E, d) and infinitely many open sets {U_{i} ⊂ E}_{i∈I} such that ∩_{i∈I}U_{i} is not open. You could consider suitable intervals in R for example.

A.2. Let X be a subset of a metric space (E, d). Show that:

\overline{X}^{∘} = \overline{\overline{X}^{∘}}^{∘} and that \overline{X^{∘}} = \overline{\overline{X^{∘}}^{∘}}.

Give an example of a subset X of R such that the following sets:

X, \overline{X}, X^{∘}, \overline{X}^{∘}, \overline{X^{∘}}, \overline{\overline{X}^{∘}}, \overline{X^{∘}}^{∘},

are pairwise distinct.

A.3. Induced topology. Let (E, d) be a metric space, and let X be a subset of E so that (X, d) is itself a topological space with the induced distance. Show that U ⊂ X is open in X if and only if there exists an open set V of E such that U = X ∩ V.

Be careful: if U ⊂ X is open in X, it is not necessarily open in E.

A.4. Product topology. Let (E_{1}, d_{1}) and (E_{2}, d_{2}) be two metric spaces. Define a distance d on E_{1} × E_{2} by

d((x_{1}, x_{2}), (y_{1}, y_{2})) = d_{1}(x_{1}, y_{1}) + d_{2}(x_{2}, y_{2}).

(1) Show that U ⊂ E_{1} × E_{2} is open if and only if for each point (u_{1}, u_{2}) ∈ U, there exist U_{1} and U_{2} open in E_{1} and E_{2} with (u_{1}, u_{2}) ∈ U_{1} × U_{2} and U_{1} × U_{2} ⊂ U.

(2) If X is another metric space, show that (f_{1}, f_{2}) : X → E_{1} × E_{2} is continuous if and only if f_{1} and f_{2} both are.

2 MATH 25

B. Problems graded by Benjamin

B.1. The p-adic topology. Given a ∈ Q^{∗} there is a unique v = v(a) ∈ Z such that one can write a = p^{v}n/d with n, d ∈ Z relatively prime to p. For x, y ∈ Q define d_{p}(x, y) = p^{−v(x−y)} if x ≠ y and d(x, x) = 0.

(1) Show that (Q, d_{p}) is a metric space and that actually d(x, z) ≤ max(d(x, y), d(y, z));

(2) show that if x ∈ B(a, r), then B(x, r) = B(a, r) so that any point of the ball B(a, r) is a “center” of that ball;

(3) show that given two balls B(a_{1}, r_{1}) and B(a_{2}, r_{2}), then either they’re disjoint or one

is contained in the other; This goes to show that metric spaces can be quite strange.

B.2. Accumulation points. If u = {u_{n}}_{n≥1} is a sequence of elements of a metric space (E, d), let A(u) be the set of its accumulation points. Show that A(u) is closed.

B.3. Let f : [0, 1] → R be defined by f(x) = \sqrt{x}. Show that f is uniformly continuous.

B.4. We say that a function f : X → Y is K-Lipshitz if d_{Y} (f(x), f(x′)) ≤ K · d_{X}(x, x′) for all x, x′ ∈ X. Check that such a function is uniformly continuous.

Show that for every 0 < a < 1, there exists K_{a} such that f(x) = \sqrt{x} is K_{a}-Lipshitz on [a, 1]. What happens if a = 0?

C. Problems graded by Inna

C.1. We say that a function f : X → Y is continuous at a point x if for every ϵ > 0, there exists δ > 0 such that if d(x, x′) < δ, then d(f(x), f(x′)) < ϵ. This way, f : X → Y is continuous if and only if it is continuous at every x ∈ X.

(1) Show that f is continuous at x if and only if for every sequence {x_{n}}_{n} which converges to x, we have f(x_{n}) → f(x);

(2) If x ∈ Q, one can write x = p/q in a unique way with gcd(p, q) = 1 and q ≥ 1 (if x = 0, set p = 0 and q = 1). Let f : [0, 1] → R be defined by

f(x) = \begin{cases}0 & if x ∈ R \ Q, \\ 1/q & if x = p/q ∈ Q. \end{cases}

At what points x of [0, 1] is f continuous?

C.2. Let R^{2} be the plane with the usual distance and let x_{1}, ⋯, x_{n} be finitely many points. Show that R^{2} \ {x_{1}, ⋯, x_{n}} is connected.

HOMEWORK 2 — DUE OCT 3RD 3

C.3. This problem has been removed and will be in the third set. C.4. Show that a circle is not homeomorphic to an interval of R. Show also that a “figure eight” is not homeomorphic to a circle.

Hint: think about connectedness.

D. Supplementary problems

If you get bored, do the following. D.1. In exercise B.1, show that (Q, d_{p}) is not complete. D.2. Show that if F is a closed subset of a metric space E and if f : F → [0, 1] is a continuous function, then there exists a continuous function f̃: E → [0, 1] which coincides with f on F.