MASTER’S EXAM: GENERAL TOPOLOGY June 1997

1. Let X = {a, b, c} with the topology {X, ∅, {a}, {b, c}}. Let A = {a, b}. Find the interior, closure, set of all limit points, and boundary of A in X.

2. Let f : X → Y be a function.

(a) Prove that if f is continuous, then for every sequence {x_{n}} such that x_{n} → x in X one has f(x_{n}) → f(x) in Y.

(b) Prove that if X satisfies the first axiom of countability, then the converse holds.

3. Prove that every infinite subset of a compact space has a limit point.

4. Prove that every metrizable space is normal.

5. Consider the circle S^{1}, the line R, the non-negative ray [0, +∞), and the non-positive ray (−∞, 0]. For each pair of these spaces determine whether or not they are homeomorphic. Give brief proofs of your answers.