MASTER’S EXAM - TOPOLOGY - JANUARY 1998

1. Prove that X is Hausdorff if and only if the diagonal Δ = {(x, x)|x ∈ X} is closed in X × X.

2. Let {C_{n}} be a sequence of closed, non-empty subsets of a compact space X such that C_{n} ⊇ C_{n+1} for all n. Prove from the definition of compactness that ∩C_{n} ≠ Ø.

3. Let U be a connected, non-empty open subset of R^{2}. Prove that U is path-connected.

4. Let f : X → Y be a continuous bijection which is a closed map. Prove that f is a homeomorphism.

5. Prove that every separable metric space satisfies the second axiom of countability.