Book:Jorge Picado/Frames and Locales

Jorge Picado and Aleš Pultr: Frames and Locales

Published $\text {2012}$, Birkhäuser

ISBN 978-3-0348-0153-9

Subject Matter

• Topology

Contents

Preface

Introduction

I. Spaces and Lattices of Open Sets

1.$\quad$Sober Spaces

2.$\quad$The axiom $T_D$: another case of spaces easy to reconstruct

3.$\quad$Summing up

4.$\quad$Aside: several technical properties of $T_D$-spaces

II. Frames and Locales. Spectra

1.$\quad$Frames

2.$\quad$Locales and localic maps

3.$\quad$Points

4.$\quad$Spectra

5.$\quad$The unit $\sigma$ and spatiality

6.$\quad$The unit $\lambda$ and sobriety

III. Sublocales

1.$\quad$Extremal monomorphisms in Loc

2.$\quad$Sublocales

3.$\quad$The co-frame of sublocales

4.$\quad$Images and preimages

5.$\quad$Alternative representations of sublocales

6.$\quad$Open and closed sublocales

7.$\quad$Open and closed localic maps

8.$\quad$Closure

9.$\quad$Preimage as a homomorphism

10.$\,\,\,$ Other special sublocales: one-point sublocales and Boolean ones

11.$\,\,\,$ Sublocales as quotients. Factorizing frames is surprisingly easy

IV. Structure of Localic Morphisms. The Categories Loc and Frm

1.$\quad$Special morphisms. Factorizing in Loc and Frm

2.$\quad$The down-set functor and free constructions

3.$\quad$Limits and a colimit in Frm

4.$\quad$Coproducts of frames

5.$\quad$More on the structure of coproduct

6.$\quad$Epimorphisms in Frm

V. Separation Axioms

1.$\quad$Instead of $T_1$: subfit and fit

2.$\quad$Mimicking the Hausdorff axiom

3.$\quad$I-Haudorff frames and regular monomorphisms

4.$\quad$Aside: Raney identity

5.$\quad$Quite like the classical case: Regular, completely regular and normal

6.$\quad$The categories RegLoc,CRegLoc, HausLoc, FitLoc

VI. More on Sublocales

1.$\quad$Subspaces and sublocales of spaces

2.$\quad$Spatial and induced sublocales

3.$\quad$Complemented sublocales of spaces are spatial

4.$\quad$The zero-dimensionlity of $\map {\mathscr {Sl}} L^{op}$ and a few consequences

5.$\quad$Difference and pseudodifference, residua

6.$\quad$Isbell's Development Theorem

7.$\quad$Locales with no non-spatial sublocales

8.$\quad$Spaces with no non-induced sublocales

VII. Compactness and Local Compactness

1.$\quad$Basics, and a technical lemma

2.$\quad$Compactness and separation

3.$\quad$Kuratowski-Mrówka characterization

4.$\quad$Compactification

5.$\quad$Well below and rather below. Continuous completely regular frames

6.$\quad$Continuous is the same as locally compact. Hofmann-Lawson duality

7.$\quad$One more spatiality theorem

8.$\quad$Supercompactness. Algrbraic, superalgebraic and supercontinuous frames

VIII. (Symmetric) Uniformity and Nearness

1.$\quad$Background

2.$\quad$Uniformity and nearness in the point-free context

3.$\quad$Uniform homomorphisms. Modelling embeddings. Products

4.$\quad$Aside: admitting nearness in a weaker sense

5.$\quad$Compact uniform and nearness frames. Finite covers

6.$\quad$Completeness and completion

7.$\quad$Functoriality. CUniFrm is corefllective in UniFrm

8.$\quad$An easy completeness criterion

IX. Paracompactness

1.$\quad$Full normality

2.$\quad$Paracompactness, and its various guises

3.$\quad$An elegant, specifically point-free, characterization of paracompactness

4.$\quad$A pleasant surprise: paracompact (co)reflection

X. More about Completion

1.$\quad$A variant of the completion of uniform frames

2.$\quad$Two applications

3.$\quad$Cauchy points and the resulting space

4.$\quad$Cauchy spectrum

5.$\quad$Cauchy completion. The case of countably generated uniformities

6.$\quad$Generalized Cauchy points

XI. Metric Frames

1.$\quad$Diameters and metric diameters

2.$\quad$Metric spectrum

3.$\quad$Uniform Metrization Theorem

4.$\quad$Metrization theorems for plain frames

5.$\quad$Categories of metric frames

XII. Entourages. Asymmetric Uniformity

1.$\quad$Entourages

2.$\quad$Untiformities via entourages

3.$\quad$Entourages versus covers

4.$\quad$Asymmetric unitformity: the classical case

5.$\quad$Biframes

6.$\quad$Quasi-uniformity in the point-free context via paircovers

7.$\quad$The adjunction QUnif $\rightleftarrows$ QUniFrm

8.$\quad$Quasi-uniformity in the point-free context via entourages

XIII. Connectedness

1.$\quad$A few observations about sublocales

2.$\quad$Connected and disconnected locales

3.$\quad$Locally connected locales

4.$\quad$A weird example

5.$\quad$A few notes

XIV. The Frame of Reals and Real Functions

1.$\quad$The frame $\map {\mathfrak L} \R$ of reals

2.$\quad$Properties of $\map {\mathfrak L} \R$

3.$\quad$$\map {\mathfrak L} \R$ versus the usual space of reals

4.$\quad$The metric uniformity of $\map {\mathfrak L} \R$

5.$\quad$Continuous real functions

6.$\quad$Cozero elements

7.$\quad$More general real functions

8.$\quad$Notes

XV. Localic Groups

1.$\quad$Basics

2.$\quad$The category of localic groups

3.$\quad$Closed Subgroup Theorem

4.$\quad$The multiplication $\mu$ is open. The semigroup of open parts

5.$\quad$Uniformities

6.$\quad$Notes

Appendix I. Posets

1.$\quad$Basics

2.$\quad$Zorn's Lemma

3.$\quad$Suprema and infima

4.$\quad$Semilatices, lattices and complete lattices. Completion

5.$\quad$Galois connections (adjunctions)

6.$\quad$(Semi)lattices as algrbras. Distributive lattices

7.$\quad$Pseudocomplements and complements. Heyting and Boolean algebras

Appendix II. Categories

1.$\quad$Categories

2.$\quad$Functors and natural transformations

3.$\quad$Some basic constructions

4.$\quad$More special morphisms. Factorization

5.$\quad$Limits and colimits

6.$\quad$Adjunction

7.$\quad$Adjointness and (co)limits

8.$\quad$Reflective and coreflective subcategories

9.$\quad$Monads

10.$\,\,\,$ Algebras in a category

Bibliography

List of Symbols

List of Categories

Index