# Wallman compactification

*Wallman–Shanin compactification, , of a topological space satisfying the axiom (cf. Separation axiom)*

The space whose points are maximal centred systems of closed sets in (cf. Centred family of sets). The topology in is given by the closed base , where ranges over all closed sets in and consists of precisely those for which for some .

This compactification was described by H. Wallman [1].

The Wallman compactification is always a compact -space; for a normal space it coincides with the Stone–Čech compactification.

If in defining the extension one chooses not all closed sets, but only those contained in a certain fixed closed base, one obtains a so-called compactification of Wallman type. Not every Hausdorff compactification of a Tikhonov space is a compactification of Wallman type.

#### References

[1] | H. Wallman, "Lattices and topological spaces" Ann of Math. , 39 (1938) pp. 112–126 |

#### Comments

Compactifications that are not Wallman compactifications were constructed by V.M. Ul'yanov [a1].

#### References

[a1] | V.M. Ul'yanov, "Solution of a basic problem on compactifications of Wallman type" Soviet Math. Dokl. , 18 (1977) pp. 567–571 Dokl. Akad. Nauk SSSR , 233 : 6 (1977) pp. 1056–1059 |

[a2] | R.A. Alo, H.L. Shapiro, "Normal bases and compactifications" Math. Ann. , 175 (1968) pp. 337–340 |

[a3] | O. Frink, "Compactifications and semi-normal spaces" Amer. J. Math. , 86 (1964) pp. 602–607 |

[a4] | R.C. Walker, "The Stone–Čech compactification" , Springer (1974) |

**How to Cite This Entry:**

Wallman compactification.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Wallman_compactification&oldid=15108