Solution
= Solution
The map $f:X\to Y$ is continuous if
$$
\boxed{U\text{ open in }Y\ \Longrightarrow\ f^{-1}(U)\text{ open in }X.}
$$
This condition concerns inverse images of open sets; images of open sets need not be open.
= Solution
The map $f:X\to Y$ is continuous if
$$
\boxed{U\text{ open in }Y\ \Longrightarrow\ f^{-1}(U)\text{ open in }X.}
$$
This condition concerns inverse images of open sets; images of open sets need not be open.