Solution
= Solution
Let $(V_i)$ be an <open cover> of $Y$. By <continuity>, the <inverse images> $f^{-1}(V_i)$ are <open> in $X$; by <surjectivity>, they cover $X$. <Compactness> supplies finitely many covering $X$. Applying $f$ and using $f(X)=Y$ shows that the corresponding finitely many $V_i$ cover $Y$.
Thus \b[a continuous <surjective function> from a <compact space> has <compact> target]. This is the <continuous image of a compact space> theorem.