PMTH331 (2017) TOPOLOGY (Due Date: 28th March) ASSIGNMENT 2 Question 1. Let (X; %) be a metric space. Prove that %: X × X −! R+ 0 ; (x; y) −! %(x; y) 1 + %(x; y) is also a metric on X. Let (Y; σ) be any metric space. Take functions f : X −! Y and g: Y −! X. Prove that (a) f is continuous with respect to % if and only if it is continuous with respect to %; (b) g is continuous with respect to % if and only if it is continuous with respect to %: Question 2. Let (X; %) be a metric space. Taking R with its Euclidean metric, , and X × X with one of the canonical metrics on the Cartesian product. Prove that %: X × X −! R+ 0 ; (x; y) 7−! %(x; y) is continuous. Given metric spaces, (X; %) and (Y; σ), the function f : X −! Y is uniformly continuous if and only if given any " > 0 there is a δ > 0 with σf(u); f(v) < " whenever %(u; v) < δ. Is %: X × X ! R+ 0 uniformly continuous with respect to the Euclidean metric on R and the product metric on X × X? Question 3. Let  be the Euclidean metric on Rn and let A = [aij]n×n be a positive definite symmetric real matrix. Define %: Rn × Rn −! R+ 0 ; (x; y) 7−! vuut nX i;j=1 aij(xi − yi)(xj − yj) where x = (x1; : : : ; xn) and y = (y1; : : : ; yn). (a) Show that (Rn; %) and (Rn; ) are isometric metric spaces. (b) Show that the function F : (Rn; ) −! (Rn; %); x 7−! x is continuous. Question 4. (i) Show that the complement of any finite subset of a metric space is open. (ii) Show that every subset of a metric space is open if and only if each singleton subset is open. Question 5*. [This question is compulsory for Pmth431 students but optional for Pmth331 students.](a) Let %: X × X ! R+ be a metric on X and let φ: R+ ! R+ be a function such that 1. φ(0) = 0 2. φ is strictly increasing, i.e. for x1 < x2 we have φ(x1) < φ(x2) 3. φ is concave, i.e. for 0 ≤ α ≤ 1 and x1 < x2 we have φ(αx1 + (1 − α)x2) ≥ αφ(x1) + (1 − α)φ(x2): Show that φ ◦ % is a metric. [Hint. Show that φ is sub-additive, that is φ(a + b) ≤ φ(a) + φ(b) for a; b ≥ 0.] (b) Let %i: X × X ! R+ 0 with i 2 N be a family of metrics on X. Show that % ¯ = 1 X i =0 1 i2 %i 1 + %i is a metric on X. 5