On the topological completion

Howard Curzer, Anthony W. Hager

Research output: Contribution to journalArticle

6 Scopus citations

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 languageEnglish
Pages (from-to)365-370
Number of pages6
JournalProceedings of the American Mathematical Society
Volume56
Issue number1
DOIs
StatePublished - Apr 1976

Keywords

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

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