### Abstract

Let A be a Tychonoff space. As is well known, the points of the Stone-Cech compactification βX “are” the zero-set ultrafilters of X, and the points of the Hewitt real-compactification νX are the zero-set ultrafilters which are closed under countable intersection. It is shown here that a zeroset ultrafilter is a point of the Dieudonna topological completion δX iff the family of complementary cozero sets is σ-discretely, or locally finitely, additive. From this follows a characterization of those dense embeddings X ⊂ Y such that each continuous metric space-valued function on X extends over Y, and a somewhat novel proof of the Katetov-Shirota Theorem.

Original language | English |
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Pages (from-to) | 365-370 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 56 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1976 |

### Keywords

- Extension of functions
- Katetov-Shirota Theorem
- Locally finite
- Topological completion
- Zero-set ultrafilter
- σ-discrete

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## Cite this

Curzer, H., & Hager, A. W. (1976). On the topological completion.

*Proceedings of the American Mathematical Society*,*56*(1), 365-370. https://doi.org/10.1090/S0002-9939-1976-0415573-9