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).