Q1. (Furstenberg) Let a, b ∈ Z with a ≠ 0 and define

Aa,b = {an + b | n ∈ Z} = aZ + b.

(a) Show that A = {A_a,b |a, b ∈ Z, a ≠ 0} is a basis for a topology on Z (called the arithmetic progression topology).

(b) Show that with the arithmetic progression topology, finite subsets of Z are not open.

(c) Show that A_a,b is both open and closed.

(d) Deduce that there are infinitely many primes.

Q2. Let X and Y be topological spaces, A ⊆ X, and f: X → Y be a homeomorphism (so X and Y are homeomorphic).

(a) Prove or disprove: (f(A))^- is homeomorphic to A^-.

(b) Prove or disprove: Int(A) is homeomorphic to Int(f(A)).

(c) Let ∂A = A^-\ Int(A) denote the boundary of A. Prove or disprove: Then ∂A is homeomorphic to ∂f(A).

(d) Show that R² is homeomorphic to B₁(0, 0), the open ball of radius one centered at the origin in the plane. Observe that ∂R² = ∅ and ∂B₁(0, 0) ≅ S¹. Explain why this example does or does not contradict your results in parts (a), (b), and (c).

Q3. Suppose f: X → X is a continuous bijection. Prove or disprove: f is a homeomorphism.

Q4. Show that R²\ {0} is homeomorphic to R × S¹.

Q5. Prove or disprove: X is Hausdorff where X is

(a) Z with the digital line topology.

(b) Z × Z with the digital plane topology.

(c) Z with the arithmetic progression topology.