Suppose that \(X\) is compact and nonempty, and \(Y\) is Hausdorff and connected. Prove that any continuous, open map \(f:X\to Y\) is onto.
Solution
1. \(f(X)\) is closed in \(Y\).
\(f(X)\) is compact as a continuous image of a compact space, and compact subsets of the Hausdorff space \(Y\) are closed.
2. \(f(X)\) is open in \(Y\).
\(X\) is open in itself and \(f\) is an open map.
3. Q.E.D.
\(f(X)\) is nonempty because \(X\) is, and by steps 1 and 2 it is clopen in the connected space \(Y\), so \(f(X)=Y\).