1. Let $\Z$ be equipped with the induced topology from $\R$, and $Y$ any topological space. Prove that any map $f: \Z \to Y$ is continuous.

2. Let $X = [0, 2\pi ) \subset \R$ be equipped with the induced topology from $\R$, and $Y \In \R^2$ be the unit circle. Let $f: X \to Y$ be given by $f(t) = (\cos(t), \sin(t))$. Prove that $f$ is a bijection and continuous, but $f^{-1}: Y \to X$ is not continuous.

3. Let $f: X \to Y$ be a continuous map. Let $B \subset Y$ be a subset. Are the following statements true? Please give proofs or counter-examples.

• If $B$ is closed in $Y$, then $f^{-1}(B)$ is closed in $X$.
• If $B$ is compact in $Y$, then $f^{-1}(B)$ is compact in $X$.

4. Let $\Q \subset \R$ be equipped with the induced topology from $\R$. Let $\N$ be equipped with the discrete topology. Can you find a continuous map $f: \Q \to \N$ such that $f$ is a bijection? If yes, give a construction, if no, give a proof.

5. Let $X = [0,1] \subset \R$, and let $Y = \{0, 1\}$, both with the induced topology from $\R$. Is there a continuous map from $Y$ to $X$? Is there a continuous map from $X$ to $Y$? Explain your answer.