Closed mapping

A mapping of one topological space to another, under which the image of every closed set is a closed set. The class of continuous closed mappings plays an important role in general topology and its applications. Continuous closed compact mappings are called perfect mappings. A continuous mapping , , of -spaces is closed if and only if the decomposition is continuous in the sense of Aleksandrov (upper continuous) or if for every open set in , the set is open in . The latter property is basic to the definition of upper semi-continuous many-valued mappings. That is, is closed if and only if its (many-valued) inverse mapping is upper continuous. Any continuous mapping of a Hausdorff compactum onto a Hausdorff space is closed. Any continuous closed mapping of -spaces is a quotient mapping; the converse is false. The orthogonal projection of a plane onto a straight line is continuous and open, but not closed. Similarly, not every continuous closed mapping is open. If is continuous and closed, with and completely regular, then for any point . (Here is the Stone–Čech compactification and is the continuous extension of the mapping to the Stone–Čech compactifications of and ); the converse is true in the class of normal spaces. Passage to the image under a continuous closed mapping preserves the following topological properties: normality; collection-wise normality; perfect normality; paracompactness; weak paracompactness. Complete regularity and strong paracompactness need not be preserved under continuous closed — and even perfect — mappings. Passage to the pre-image under a continuous closed mapping need not preserve the above-mentioned properties. The explanation for this is that the pre-image of a point under a continuous closed mapping need not be compact, though in many cases there is little difference between continuous closed and perfect mappings. If is a continuous closed mapping of a metric space onto a space satisfying the first axiom of countability, then is metrizable and the boundary of the pre-image is compact for every . If is a continuous closed mapping of a metric space onto a -space , then the set of all points for which is not compact is -discrete.

References

[1]	A.V. Arkhangel'skii, "Mappings and spaces" Russian Math. Surveys , 21 : 4 (1966) pp. 115–126 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 133–184
[2]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[3]	R. Engelking, "General topology" , PWN (1977) (Translated from Polish)

Comments

The notion of a closed mapping leads to the notion of an upper semi-continuous decomposition of a space. This is a decomposition of a space such that the quotient mapping is closed.

In the Russian literature denotes the closure of the set , so in this article is the closure of the fibre in the space (see also Closure of a set).