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Weil Entourages in Pointfree Topology

Ph. D. Thesis

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Jorge Picado

Departamento de Matemática

Universidade de Coimbra

1995

**Abstract: **Uniform structures for frames and their generalizations
(quasi-uniformities and nearnesses) are the subject of this thesis.
Weil's notion on entourage is extended to this framework and it is
proved that this is a basic concept on which that structures may be
axiomatized. On the other hand, it is shown that uniform frames may
also be described by gauge structures, that is, certain families of
metric diameters.

**Contents:**

INTRODUCTION

0 - PRELIMINARIES

1. Frames and topological spaces.

2. Biframes and bitopological spaces.

3. Quotients of frames.

4. Down-sets and filters.

5. Binary coproducts of frames.

6. Galois connections.

I - WEIL UNIFORM FRAMES

1. Uniform spaces.

2. Covering uniform frames.

3. Entourage uniform frames.

4. Weil uniform frames.

5. The isomorphism between the categories UFrm, WUFrm and EUFrm.

6. An application: a theorem of Efremovic for uniform spaces in pointfree context.

II - UNIFORM FRAMES IN THE SENSE OF BOURBAKI

1. Gauge spaces.

2. Metric frames.

3. Gauge frames.

4. An application: the category UFrm is fully embeddable in
a final completion of the category MFrm.

III - WEIL QUASI-UNIFORM FRAMES

1. Quasi-uniform spaces.

2. Covering quasi-uniform frames.

3. Weil quasi-uniform frames.

4. The isomorphism between the categories QUFrm and
WQUFrm.

IV - WEIL NEARNESS SPACES AND FRAMES

1. Nearness spaces.

2. Covering nearness frames.

3. Weil nearness frames.

4. Weil nearness spaces.

5. The category Wnear as a unified theory of (symmetric)
topology and uniformity.

6. Proximal frames.

APPENDIX - Hierarchy of nearness structures on sets and frames

BIBLIOGRAPHY

INDEX OF CATEGORIES

INDEX OF OTHER SYMBOLS

INDEX OF DEFINITIONS