Spring 2010 Problem 1Let (X, d) be a complete metric space, \overline{x} ∈ X and r > 0. Set D := \{x ∈ X : d(x, \overline{x}) ≤ r\}, and let f : D → X satisfying

d(f(x), f(y)) ≤ kd(x, y)

for any x, y ∈ D, where k ∈ (0, 1) is a constant. Prove that if d(\overline{x}, f(\overline{x}) ≤ r(1 − k) then f admits a unique fixed point.

Spring 2010 Problem 6 Let f : [0, 1] → ℝ. Show that f is continuous if and only if the graph of f is compact in ℝ^{2}.

Fall 2011 Problem 1Let (X, d) be a metric space and let (x_{n}) be a sequence in X. Call x ∈ X a cluster point o (x_{n}) iff there is a subsequence (x_{n_{k}})_{k≤0} with limit x.

(1) Let (a_{n}) be a sequence of distinct points in X. Construct a sequence in X for which every a_{k} is a cluster point.

(2) Can a sequence in a metric space have an uncountable number of cluster points? Prove your answer.

Fall 2017 Problem 1 Prove that every metric subspace of a separable metric space is separable.

Winter 2008 Problem 2:Consider X = ℝ^{2} with the Euclidean metric e. Define d : X × X → ℝ by d(x, y) = e(x, y) if x and y lie on the sae ray through te origin and d(x, y) = e(x, 0) + e(0, y) otherwise.

(1) Prove that (X, d) is a metric space.

(2) Give an example of a set which is open in (X, d) but not in (X, e).