For the next four problems, we let X and Y be two topological spaces and let...

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For the next four problems, we let X and Y be two topological spaces and let f : X → Y be a continuous map. Recall that for any subset AC X, ƒ(A) := {f(x) | x € A}, and for any subset BCY, f-¹(B) := {x € X | ƒ(x) ¤ B}. Problem 7. Prove that if C is a closed subset of Y, then f-¹(C) is a closed subset of X. (Hint: f-¹(C)= (ƒ-¹(C))c.) Problem 8. Prove that if K is a compact subset of X, then f(K) is a compact subset of Y. (Hint: if {Uifier is an open cover of f(K), then {f−¹(U₁)}i≤1 is an open cover of K.) Problem 9. Prove that if E is a connected subset of X, then f(E) is a connected subset of Y. (Hint: prove the contrapositive.) Problem 10. Suppose f: X→ Y is surjective. Prove that if D is a dense subset of X, then f(D) is a dense subset of Y. For the next four problems, we let X and Y be two topological spaces and let f : X → Y be a continuous map. Recall that for any subset AC X, ƒ(A) := {f(x) | x € A}, and for any subset BCY, f-¹(B) := {x € X | ƒ(x) ¤ B}. Problem 7. Prove that if C is a closed subset of Y, then f-¹(C) is a closed subset of X. (Hint: f-¹(C)= (ƒ-¹(C))c.) Problem 8. Prove that if K is a compact subset of X, then f(K) is a compact subset of Y. (Hint: if {Uifier is an open cover of f(K), then {f−¹(U₁)}i≤1 is an open cover of K.) Problem 9. Prove that if E is a connected subset of X, then f(E) is a connected subset of Y. (Hint: prove the contrapositive.) Problem 10. Suppose f: X→ Y is surjective. Prove that if D is a dense subset of X, then f(D) is a dense subset of Y.