MASTER’S EXAM - TOPOLOGY - JANUARY 2000

1. Prove that every closed subspace of a compact topological space is compact.

2. Let B be a countable basis for the topology T on the set X. Let C be an arbitrary basis for T . Prove that C has a countable subset D which is a basis for T . (Hint: Suppose B = {B_{n}}_{n∈Z_{+}}. Consider the set of those pairs of indices (i, j) such that there exists a C ∈ C with B_{i} ⊆ C ⊆ B_{j}.)

3. Prove that every connected normal space having at least two points is uncountable. (Hint: Urysohn Lemma.)

4. Let (X, d) be a metric space. Let x_{0} ∈ X. Let U = {x ∈ X | d(x, x_{0}) > 1}. Prove that U is open in X.

5. Let (X, d) be a compact metric space. Recall that if A is any covering of X by open sets, then there is a number λ > 0, called the Lebesgue number of the covering, such that if C is any set of diameter less than λ, then there exists an element A of A such that C ⊆ A. Use this fact to prove that if f : X → R is continuous, then f is uniformly continuous.