Master’s Exam Topology August 1995

1. (a) If X is a space and A ⊆ X, define the closure of A in X.

(b) Let Y be a subspace of X; let A be a subset of Y; let Ā be the closure of A in X. Prove that the closure of A in Y equals Ā ∩ Y.

2. Let I = [0, 1]. Let I^{ω} be the standard cube in ℝ^{ω} in the product topology.

(a) Prove that there is a continuous function f : I^{ω} → [0, 1] that takes the value 0 at the origin (0, 0, ...) and that takes the value 0.9 on the whole subcube [0.9, 1]^{ω}

(b) Take some f that satisfies part (a) and assume that it never takes a value < 0 or ≥ 1. Prove that there is an a < 1 such that the image of f lies in [0, a].

(c) In the situation of part (b), prove that a/2 is in the image of f.

Be sure to verify the hypothesis of any theorems or lemmas you use.

3. Give an example, with proof, showing that the quotient of a Hausdorff space need not be Hausdorff.

4. (a) Define the terms Lindelöf and sequentially compact.

(b) Prove that a space that’s Lindelöf and sequentially compact must be compact.