A characterization of closed maps using the Whyburn construction (Q1062304)

Let \(f: X\to Y\) be continuous and let X and Y be Hausdorff. Whyburn defined the unified space Z to be the disjoint union of X and Y with a set Q open in Z if and only if \(Q\cap X\) is open in X, \(Q\cap Y\) is open in Y, and for any compact \(K\subset Q\cap Y\), \(f^{-1}(K)-Q\) is compact. The author modifies the topology on \(X\cup Y\) by replacing the compact sets by single points and denote the modified Whyburn space by W. It is obvious that any set open in Z is open in W. It is shown that if f is closed, the topologies are in fact the same and if Y is first countable, then Z and W being the same implies that f is closed. This yields the following corollary: Any continuous function from a Hausdorff space into the reals (or any metric space) is closed if and only if the Whyburn space and the modified Whyburn space are the same.