Weil Entourages in Pointfree Topology
Ph. D. Thesis
Departamento de Matemática
Universidade de Coimbra
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
0 - PRELIMINARIES
1. Frames and topological spaces.
I - WEIL UNIFORM FRAMES
2. Biframes and bitopological spaces.
3. Quotients of frames.
4. Down-sets and filters.
5. Binary coproducts of frames.
6. Galois connections.
1. Uniform spaces.
II - UNIFORM FRAMES IN THE SENSE OF BOURBAKI
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.
1. Gauge spaces.
III - WEIL QUASI-UNIFORM FRAMES
2. Metric frames.
3. Gauge frames.
4. An application: the category UFrm is fully embeddable in
a final completion of the category MFrm.
1. Quasi-uniform spaces.
IV - WEIL NEARNESS SPACES AND FRAMES
2. Covering quasi-uniform frames.
3. Weil quasi-uniform frames.
4. The isomorphism between the categories QUFrm and
1. Nearness spaces.
APPENDIX - Hierarchy of nearness structures on sets and frames
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.
INDEX OF CATEGORIES
INDEX OF OTHER SYMBOLS
INDEX OF DEFINITIONS