# Freudenthal compactification

A maximal compactification with zero-dimensional remainder (cf. Remainder of a space). Every peripherically-compact space has a Freudenthal compactification (this was proved by H. Freudenthal ). Among all such extensions there is a unique maximal one, and it is called the Freudenthal compactification (sometimes a Freudenthal compactification just means any compactification with zero-dimensional remainder). The Freudenthal compactification can be characterized as the (unique) perfect compactification with zero-dimensional remainder, and also as the minimal perfect compactification (see ).

#### References

[1a] | H. Freudenthal, "Neuaufbau der Endentheorie" Ann. of Math. , 43 : 2 (1942) pp. 261–279 |

[1b] | H. Freudenthal, "Kompaktisierungen und Bikompaktisierungen" Indag. Math. , 13 (1951) pp. 184–192 |

[2] | E.G. Sklyarenko, "Bicompact extensions of semibicompact spaces" Dokl. Akad. Nauk SSSR , 120 : 6 (1958) pp. 1200–1203 (In Russian) |

[3a] | E.G. Sklyarenko, "Some questions in the theory of bicompactifications" Izv. Akad. Nauk SSSR Ser. Mat. , 26 : 3 (1962) pp. 427–452 (In Russian) |

[3b] | E.G. Sklyarenko, "Bicompactifications with punctiform boundary and their cohomology groups" Izv. Akad. Nauk SSSR Ser. Mat. , 27 : 5 (1963) pp. 1165–1180 (In Russian) |

#### Comments

See also Compact extension; Compactification.

The Freudenthal compactification is used in the theory of topological groups (cf. Topological group).

**How to Cite This Entry:**

Freudenthal compactification. I.G. Koshevnikova (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Freudenthal_compactification&oldid=14855