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Homework 7. Due Friday, November 27. 1. Show that the unit circle C = {(x, y) ∈ R2 : x2 + y 2 = 1} is connected. Hint: Use polar coordinates and Theorem 2.15. 2. Let U = {(x, y) ∈ R2 : xy 6= 0}. How many connected components does this set have? Hint: One of the components is (0, ∞) × (0, ∞). 3. Show that there is no surjective continuous function f : A → B, if A = (0, 3) ∪ (3, 6), B = (0, 1) ∪ (1, 2) ∪ (2, 3). Hint: Look at the restriction of f on each subinterval. You will need to use Theorem 2.11 and also some parts of Theorem 2.15. 4. Let (X, T ) be a topological space and suppose A1 , A2 , . . . , An are connected subsets of X such that Ak ∩ Ak+1 is nonempty for each k. Show that the union of these sets is connected. Hint: Use induction.