MASTER’S EXAM - TOPOLOGY - JUNE 1999

1. Let A and B be subsets of a topological space X. For each of the following inclusions give either a proof or a counterexample.

(a) \overline{A ∩ B} ⊆ \overline{A} ∩ \overline{B}

(b) \overline{A} ∩ \overline{B} ⊆ \overline{A ∩ B}

2. Let A, X, and Y be topological spaces. Give X × Y the product topology. Define the projections π_{X} : X × Y → X and π_{Y} : X × Y → Y by π_{X}(x, y) = x and π_{Y} (x, y) = y. Prove that a function f : A → X × Y is continuous if and only if π_{X} ∘ f and π_{Y} ∘ f are continuous.

3. Give R^{n} the square metric ρ(x, y) = max{|x_{i} − y_{i}|, 1 ≤ i ≤ n}. Let A be a closed subset of R^{n} which is bounded with respect to ρ. Prove that A is compact.

4. Let A and C be subsets of a topological space X. Give C the subspace topology. Prove that if C is connected, C ∩ A ≠ ∅, and C ∩ (X − A) ≠ ∅, then C ∩ Bd A ≠ ∅. (If you wish, you may use any lemmas you like about Int A and Bd A, but be sure to explicitly state them.)

5. Let R_{ℓ} denote the real numbers in the lower limit topology. Prove that R_{ℓ} is separable.