View source for Compact mapping

A mapping of one space into another under which the pre-image of each point is compact (cf. [[Compact space|Compact space]]). The requirement of compactness is especially useful in combination with other restrictions on the mapping. The first to be distinguished are open compact mappings (cf. [[Open mapping|Open mapping]]), perfect mappings (cf. [[Perfect mapping|Perfect mapping]]), and quotient compact mappings (cf. [[Quotient mapping|Quotient mapping]]). An important special case of a compact mapping is a [[Finite-to-one mapping|finite-to-one mapping]]. Topological properties are stable most often with respect to perfect mappings, which are the most natural analogues of continuous mappings of compacta in the class of all Hausdorff spaces. A product of compact mappings is a compact mapping.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Some basic directions in general topology"  ''Russian Math. Surveys'' , '''19''' :  6  (1964)  pp. 1–39  ''Uspekhi Mat. Nauk'' , '''19''' :  6  (1964)  pp. 3–46</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,   "Mappings and spaces"  ''Russian Math. Surveys'' , '''21''' :  4  (1966)  pp. 115–162  ''Uspekhi Mat. Nauk'' , '''21''' :  4  (1966)  pp. 133–184</TD></TR></table>



====Comments====
Some mathematicians (especially topologists) use the word  "mapping"  in the sense of  "continuous mapping" . This is also the case in this article.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.K. Burke,   "Closed mappings"  G.M. Reed (ed.) , ''Surveys in general topology'' , Acad. Press  (1980)  pp. 1–32</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>