A map \(f:X\to Y\) is called closed if \(f(F)\) is closed in \(Y\) for every closed subset \(F\subseteq X\).
Suppose \(f:X\to Y\) is closed and \(f(X)\subseteq B\subseteq Y\). Show that the corestriction \[ f^B:X\to B \] is a closed map when \(B\) has the subspace topology.
Let \(\{X_\alpha\}_{\alpha\in I}\) be topological spaces and give \(\prod_{\alpha\in I}X_\alpha\) the product topology. Show by example that a coordinate projection \[ \pi_\beta:\prod_{\alpha\in I}X_\alpha\to X_\beta \] need not be a closed map.