Reference no: EM131103342
Q1. (Furstenberg) Let a, b ∈ Z with a ≠ 0 and define
Aa,b = {an + b | n ∈ Z} = aZ + b.
(a) Show that A = {Aa,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 Aa,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.
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