# Compact mapping

A mapping of one space into another under which the pre-image of each point is compact (cf. 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), perfect mappings (cf. Perfect mapping), and quotient compact mappings (cf. Quotient mapping). An important special case of a compact mapping is a 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

[1] | 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 |

[2] | 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 |

#### Comments

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

#### References

[a1] | D.K. Burke, "Closed mappings" G.M. Reed (ed.) , Surveys in general topology , Acad. Press (1980) pp. 1–32 |

[a2] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |

**How to Cite This Entry:**

Compact mapping. A.V. Arkhangel'skii (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Compact_mapping&oldid=11412