Completion and Samuel compactification of nearness and uniform frames

Samuel compactification and completion of uniform frames

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410006895x ◽

1990 ◽

Vol 108 (1) ◽

pp. 63-78 ◽

Cited By ~ 38

Author(s):

B. Banaschewski ◽

A. Pultr

Keyword(s):

Uniform Space ◽

The Other ◽

Uniform Spaces ◽

Samuel Compactification ◽

Uniform Frame ◽

The One ◽

Uniform Frames

The aim of this paper is twofold: first, to construct the compact regular coreflection of uniform frames, that is, the frame counterpart of the Samuel compactification of uniform spaces (Samuel [10]), and then to use this for a new description of the completion of a uniform frame, as an alternative to those previously given by Isbell [6] on the one hand and Kříž [8] on the other. In addition, we present a few further results, as well as new proofs of known ones, that are naturally connected with completions and arise particularly easily from our approach to them. Most prominently among these, we identify the uniform space of minimal Cauchy filters of a uniform frame as the spectrum of its completion.

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On completion in the category SSNσFrm

Mathematica Slovaca ◽

10.2478/s12175-012-0093-y ◽

2013 ◽

Vol 63 (2) ◽

Author(s):

Inderasan Naidoo

Keyword(s):

Similar Construction ◽

Total Boundedness ◽

Czechoslovak Math ◽

Samuel Compactification ◽

Phd Thesis ◽

Uniform Frames

AbstractWe introduce the category SSN σ Frm of super strong nearness σ-frames and show the existence of a completion for a super strong nearness σ-frame unique up to isomorphism by the similar construction presented in [WALTERS-WAYLAND, J. L.: Completeness and Nearly Fine Uniform Frames. PhD Thesis, Univ. Catholique de Louvain, 1996] and [WALTERS-WAYLAND, J. L.: A Shirota Theorem for frames, Appl. Categ. Structures 7 (1999), 271–277]. Completion is also shown to be a coreflection in SSN σ Frm. We also engage with the notion of total boundedness for nearness σ-frames and provide a characterization of the Samuel compactification of a nearness σ-frame alternative to the description in [NAIDOO, I.: Samuel compactification and uniform coreflection of nearness σ-frames, Czechoslovak Math. J. 56(131) (2006), 1229–1241].

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