C-4 - Perfect Maps

Abstract

A perfect map : X→Y is a closed, continuous, and onto map with ƒ−1(y) compact in X for every y ∈ Y. Here, the term map (or mapping) always means a continuous map. All topological spaces considered are assumed to be Hausdorff spaces. A closed map means ƒ(A) is closed in Y for every closed set A ⊆X. It is sometimes interesting to consider a continuous closed map with a condition on the fibers that is weaker than compact (such as Lindelöf) or with no condition at all. A closed map with countably compact fibers is called a quasi-perfect map. Under any perfect map : X →Y, if C ⊆Y is a compact subset of Y, then the preimage ƒ−1(C) is a compact subset of X. Thus, if ƒ: X →Y is a continuous map, with the preimage ƒ−1(C) compact in X for every compact C ⊆Y (sometimes called a proper map), and Y is a Hausdorff k-space, then ƒ is a perfect map. Because compactness is preserved under a continuous map and compact subspaces of a Hausdorff space are closed, it is clear that any continuous map from a compact space X onto a Hausdorff space Y is a perfect map.