Extending open maps to Stone-Čech compactifications
The question was answered on MathOverflow by user Bill Johnson. With his permission, I post it here as well.
Let $Y=(-1/n)_{n=1}^\infty \cup \{0\}$, $B$ the positive integers, $X=Y\cup B$ with the topology they inherit from the real line. Define $f:X\to Y$ to be the identity on $Y$ and $f(n)=-1/n$ for $n$ in $B$. The closure of $2B$ in $\beta X$ is open and onto $\{0\} \cup (1/2n)_{n=1}^\infty$ in $Y$, which is not open.