Let $X$ and $Y$ be topological spaces. What does it mean to say that a function $f: X \rightarrow Y$ is continuous?

Are the following statements true or false? Give proofs or counterexamples.

(i) Every continuous function $f: X \rightarrow Y$ is an open map, i.e. if $U$ is open in $X$ then $f(U)$ is open in $Y$.

(ii) If $f: X \rightarrow Y$ is continuous and bijective then $f$ is a homeomorphism.

(iii) If $f: X \rightarrow Y$ is continuous, open and bijective then $f$ is a homeomorphism.