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(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪ A2 ∪...

(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪ A2 ∪ ... ∪ An is countable. (Hint: Induction.)

(6) Let F be the set of all functions from R to R. Show that |F| > 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)

(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4}, {1, 4}, {1, 2, 3, 4}}, and S = {∅, {6}, {7}, {6, 7}, {5, 6, 7, 8}}. Prove that (X, T ) and (Y, S) are homeomorphic.

(8) Let X = {1, 2}, Y = {3, 4}, T = {∅, {1, 2}}, and S = {∅, {3}, {3, 4}}. Prove that (X, T ) and (Y, S) are not homeomorphic

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