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 $ f : X \rightarrow Y $, $ f ( X ) = Y $, of $ T _ {1} $- spaces is closed if and only if the decomposition $ \{ {f ^ { - 1 } y } : {y \in Y } \} $ is continuous in the sense of Aleksandrov (upper continuous) or if for every open set $ U $ in $ X $, the set $ f ^ { \# } = \{ {y \in Y } : {f ^ { - 1 } y \in U } \} $ is open in $ U $. The latter property is basic to the definition of upper semi-continuous many-valued mappings. That is, $ f $ 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 $ T _ {1} $- 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 $ f : X \rightarrow Y $ is continuous and closed, with $ X $ and $ Y $ completely regular, then $ \overline{f}\; {} ^ { - 1 } y = [ f ^ { - 1 } y ] \beta X $ for any point $ y \in Y $. (Here $ \beta X $ is the Stone–Čech compactification and $ \overline{f}\; : \beta X \rightarrow \beta Y $ is the continuous extension of the mapping to the Stone–Čech compactifications of $ X $ and $ Y $); 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 $ f $ is a continuous closed mapping of a metric space $ X $ onto a space $ Y $ satisfying the first axiom of countability, then $ Y $ is metrizable and the boundary of the pre-image $ f ^ { - 1 } y $ is compact for every $ y \in Y $. If $ f $ is a continuous closed mapping of a metric space $ X $ onto a $ T _ {1} $- space $ Y $, then the set of all points $ y \in Y $ for which $ f ^ { - 1 } y $ is not compact is $ \sigma $- 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 $ E $ of a space $ X $ such that the quotient mapping $ q: X \rightarrow X/E $ is closed.

In the Russian literature $ [ A] $ denotes the closure of the set $ A $, so in this article $ [ f ^ { - 1 } y] \beta X $ is the closure of the fibre $ f ^ { - 1 } y $ in the space $ \beta X $( see also Closure of a set).