C-3 - Closed Maps

Abstract

A continuous map : X→Y is called a closed map if for every closed set A ⊆X, the image ƒ(A) is closed in Y. The classes of T1-spaces, normal spaces, perfectly normal spaces and collection-wise normal spaces are preserved under closed maps. If X has one of these properties, and : X→Y is a closed map onto Y, then Y also has the corresponding property. The separation properties of Hausdorff, regular, and Tychonoff are not generally preserved to closed images. Some other properties preserved by closed maps are paracompactness, metacompactness onto a Hausdorff range, subparacompactness, submetacompactness, σ -spaces, Σ#-spaces, and stratifiable spaces (M3-spaces). The Tychonoff property is preserved under an open-and closed map. This can be useful in showing that if ƒ: X→Y is an open-and-closed map of a normal space X onto a space Y, then the continuous extension βƒ over the Stone–Čech compactifications βX and βY is also open-and-closed. If ƒ: X →Y is a continuous map of a normal space X onto a Tychonoff space Y and β : βX → βY is its continuous extension, then the map ƒ is closed if and only if (iff) (βƒ)−1(y) is equal to c1βXƒ−1(y) for every y ∈ Y.