In the following, all the subsets of $\R$, or $\R^2$, are endowed with the induced topology.

1. Let $\Z \subset \R$ and $Y$ any topological space. Prove that any map $f: \Z \to Y$ is continuous.

2. Let $X = [0, 2\pi ) \subset \R$, and $Y \subset \R^2$ 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. (Remark: To show $f$ is a bijection and continuous, you may consider $F:\R \to \R^2$, where $F(t) = (\cos(t), \sin(t))$. You can prove $F$ is continuous by proving each component of $F$ is continuous. )

3. Let $f: X \to Y$ be a continuous map. Let $B \subset Y$ be a subset. Are the following statements true? Please explain.

• 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. 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 $Y = \{0, 1\}$. Is there a continuous map from $Y$ to $X$? Is there a continuous map from $X$ to $Y$? Explain your answer. (Optional, is there a continuous and surjective map from $X$ to $Y$? )