1. Let \(X\) be a topological space and let \(A, B \subseteq X\) are two connected sets with nonempty intersection. Show that \(A \cup B\) is connected.

  2. Let \(X\) be a topological space and suppose \(C_1, C_2 \subseteq X\) are two closed sets satisfying that both \(C_1 \cup C_2\) and \(C_1 \cap C_2\) are connected. Show that both \(C_1\) and \(C_2\) are connected.

  3. Show that the subset of \(\mathbb{R} \times \mathbb{R}\) defined by \(\{(x, 0) \mid x \in \mathbb{R}\} \cup \{(0, x) \mid x \in \mathbb{R}\}\) is connected. Show it is metrizable as well.

  4. Exercise 3.61 in the textbook

  5. Exercise 3.62 in the textbook

  6. Exercise 3.36 in the textbook (the author uses the term “distance” to mean “metric”). Does the topology on \(Y\) induced by this metric make the space connected or disconnected?

  7. Let \(f : X \rightarrow Y\) be a continuous function between topological spaces. If \(A \subseteq X\) is connected can we conclude that \(f(A)\) is connected? If \(A \subseteq Y\) is connected can we conclude that \(f^{-1}(A)\) is connected? Prove your answers (if you say no, give a specific counterexample)