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\begin{document}

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\section*{Frames and Locales}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include mathematicscontents]]

This entry provides a hyperlinked index for the book

\begin{itemize}%
\item [[Jorge Picado]], [[Aleš Pultr]],

\emph{Frames and Locales. Topology without points},

Frontiers in Mathematics, Birkhäuser (2012)



\end{itemize}
on basics of [[locales]] and [[pointfree topology]].

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{i_spaces_and_lattices_of_open_sets}{I. Spaces and [[Lattices]] of Open Sets}\dotfill \pageref*{i_spaces_and_lattices_of_open_sets} \linebreak
\noindent\hyperlink{ii_frames_and_locales_spectra}{II. [[Frames]] and [[Locales]]. Spectra}\dotfill \pageref*{ii_frames_and_locales_spectra} \linebreak
\noindent\hyperlink{iii_sublocales}{III. [[Sublocales]]}\dotfill \pageref*{iii_sublocales} \linebreak
\noindent\hyperlink{iv_structure_of_localic_morphisms_the_categories_loc_and_frm}{IV. Structure of [[Localic Morphisms]]. The Categories [[Loc]] and [[Frm]]}\dotfill \pageref*{iv_structure_of_localic_morphisms_the_categories_loc_and_frm} \linebreak
\noindent\hyperlink{v_separation_axioms}{V. [[Separation Axioms]]}\dotfill \pageref*{v_separation_axioms} \linebreak
\noindent\hyperlink{vi_more_on_sublocales}{VI. More on [[Sublocales]]}\dotfill \pageref*{vi_more_on_sublocales} \linebreak
\noindent\hyperlink{vii_compactness_and_local_compactness}{VII. [[Compactness]] and [[Local Compactness]]}\dotfill \pageref*{vii_compactness_and_local_compactness} \linebreak
\noindent\hyperlink{viii_symmetric_uniformity_and_nearness}{VIII. (Symmetric) Uniformity and Nearness}\dotfill \pageref*{viii_symmetric_uniformity_and_nearness} \linebreak
\noindent\hyperlink{ix_paracompactness}{IX. [[Paracompactness]]}\dotfill \pageref*{ix_paracompactness} \linebreak
\noindent\hyperlink{x_more_about_completion}{X. More about Completion}\dotfill \pageref*{x_more_about_completion} \linebreak
\noindent\hyperlink{xi_metric_frames}{XI. [[Metric Frames]]}\dotfill \pageref*{xi_metric_frames} \linebreak
\noindent\hyperlink{xii_entourages_asymmetric_uniformity}{XII. [[Entourages]]. [[Asymmetric Uniformity]]}\dotfill \pageref*{xii_entourages_asymmetric_uniformity} \linebreak
\noindent\hyperlink{xiii_connectedness}{XIII. [[Connectedness]]}\dotfill \pageref*{xiii_connectedness} \linebreak
\noindent\hyperlink{xiv_frame_of_reals_and_real_functions}{XIV. Frame of Reals and Real Functions}\dotfill \pageref*{xiv_frame_of_reals_and_real_functions} \linebreak
\noindent\hyperlink{xv_localic_groups}{XV. [[Localic Groups]]}\dotfill \pageref*{xv_localic_groups} \linebreak
\noindent\hyperlink{appendix_i_posets}{Appendix I. [[Posets]]}\dotfill \pageref*{appendix_i_posets} \linebreak
\noindent\hyperlink{appendix_ii_categories}{Appendix II. Categories}\dotfill \pageref*{appendix_ii_categories} \linebreak


The following lists chapterwise linked lists of keywords to relevant and related existing entries, as far as they already exist.

\hypertarget{i_spaces_and_lattices_of_open_sets}{}\subsection*{{I. Spaces and [[Lattices]] of Open Sets}}\label{i_spaces_and_lattices_of_open_sets}

\begin{enumerate}%
\item [[Sober spaces]]
\item The axiom $T_D$: another case of spaces easy to reconstruct
\item Summing up
\item Aside: several technical properties of $T_D$-spaces

\end{enumerate}
\hypertarget{ii_frames_and_locales_spectra}{}\subsection*{{II. [[Frames]] and [[Locales]]. Spectra}}\label{ii_frames_and_locales_spectra}

\begin{enumerate}%
\item [[Frames]]
\item [[Locales]] and [[localic maps]]
\item Points
\item Spectra
\item The unit σ and [[spatiality]]
\item The unit λ and [[sobriety]]

\end{enumerate}
\hypertarget{iii_sublocales}{}\subsection*{{III. [[Sublocales]]}}\label{iii_sublocales}

\begin{enumerate}%
\item [[Extremal monomorphisms]] in [[Loc]]
\item [[Sublocales]]
\item The co-frame of [[sublocales]]
\item Images and preimages
\item Alternative representations of [[sublocales]]
\item Open and [[closed sublocales]]
\item Open and [[closed localic maps]]
\item [[Closure]]
\item Preimage as a homomorphism
\item Other special sublocales: one-point sublocales, and [[Boolean locale|Boolean]] ones
\item [[Sublocales]] as quotients. Factorizing frames is surprisingly easy

\end{enumerate}
\hypertarget{iv_structure_of_localic_morphisms_the_categories_loc_and_frm}{}\subsection*{{IV. Structure of [[Localic Morphisms]]. The Categories [[Loc]] and [[Frm]]}}\label{iv_structure_of_localic_morphisms_the_categories_loc_and_frm}

\begin{enumerate}%
\item Special morphisms. Factorizing in Loc and Frm
\item The down-set functor and free constructions
\item [[Limits]] and a [[colimit]] in [[Frm]]
\item [[Coproducts]] of [[frames]]
\item More on the structure of [[coproduct]]
\item [[Epimorphisms]] in [[Frm]]

\end{enumerate}
\hypertarget{v_separation_axioms}{}\subsection*{{V. [[Separation Axioms]]}}\label{v_separation_axioms}

\begin{enumerate}%
\item Instead of $T_1$: [[subfit]] and [[fit]]
\item Mimicking the [[Hausdorff axiom]]
\item I-Hausdorff frames and [[regular monomorphisms]]
\item Aside: Raney identity
\item Quite like the classical case: [[Regular]], [[completely regular]] and [[normal]]
\item The categories [[RegLoc]], [[CRegLoc]], [[HausLoc]] and [[FitLoc]]

\end{enumerate}
\hypertarget{vi_more_on_sublocales}{}\subsection*{{VI. More on [[Sublocales]]}}\label{vi_more_on_sublocales}

\begin{enumerate}%
\item Subspaces and [[sublocales]] of spaces
\item Spatial and induced [[sublocales]]
\item [[Complemented sublocales]] of spaces are spatial
\item The [[zero-dimensionality]] of $Sl(L)^op$ and a few consequences
\item Difference and pseudodifference, residua
\item [[Isbell’s Development Theorem]]
\item Locales with no non-spatial sublocales
\item Spaces with no non-induced sublocales

\end{enumerate}
\hypertarget{vii_compactness_and_local_compactness}{}\subsection*{{VII. [[Compactness]] and [[Local Compactness]]}}\label{vii_compactness_and_local_compactness}

\begin{enumerate}%
\item Basics, and a technical lemma
\item [[Compactness]] and [[separation]]
\item Kuratowski-Mrówka characterization
\item Compactification
\item [[Well below]] and [[rather below]]. Continuous completely regular frames
\item Continuous is the same as [[locally compact]]. [[Hofmann-Lawson duality]]
\item One more spatiality theorem
\item [[Supercompactness]]. Algebraic, superalgebraic and [[supercontinuous frames]]

\end{enumerate}
\hypertarget{viii_symmetric_uniformity_and_nearness}{}\subsection*{{VIII. (Symmetric) Uniformity and Nearness}}\label{viii_symmetric_uniformity_and_nearness}

\begin{enumerate}%
\item Background
\item [[Uniformity]] and [[nearness]] in the point-free context
\item Uniform homomorphisms. Modelling embeddings. Products
\item Aside: admitting nearness in a weaker sense
\item Compact uniform and nearness frames. Finite covers
\item [[Completeness]] and [[completion]]
\item Functoriality. [[CUniFrm]] is coreflective in [[UniFrm]]
\item An easy completeness criterion

\end{enumerate}
\hypertarget{ix_paracompactness}{}\subsection*{{IX. [[Paracompactness]]}}\label{ix_paracompactness}

\begin{enumerate}%
\item [[Full normality]]
\item [[Paracompactness]], and its various guises
\item An elegant, specifically point-free, characterization of [[paracompactness]]
\item A pleasant surprise: paracompact (co)reflection

\end{enumerate}
\hypertarget{x_more_about_completion}{}\subsection*{{X. More about Completion}}\label{x_more_about_completion}

\begin{enumerate}%
\item A variant of the [[completion of uniform frames]]
\item Two applications
\item Cauchy points and the resulting space
\item [[Cauchy spectrum]]
\item [[Cauchy completion]]. The case of countably generated uniformities
\item Generalized Cauchy points

\end{enumerate}
\hypertarget{xi_metric_frames}{}\subsection*{{XI. [[Metric Frames]]}}\label{xi_metric_frames}

\begin{enumerate}%
\item Diameters and metric diameters
\item Metric spectrum
\item [[Uniform Metrization Theorem]]
\item [[Metrization theorems]] for plain frames
\item Categories of [[metric frames]]

\end{enumerate}
\hypertarget{xii_entourages_asymmetric_uniformity}{}\subsection*{{XII. [[Entourages]]. [[Asymmetric Uniformity]]}}\label{xii_entourages_asymmetric_uniformity}

\begin{enumerate}%
\item [[Entourages]]
\item [[Uniformities]] via [[entourages]]
\item [[Entourages]] versus [[covers]]
\item [[Asymmetric uniformity]]: the classical case
\item [[Biframes]]
\item [[Quasi-uniformity]] in the point-free context via [[paircovers]]
\item The adjunction $QUnif \leftrightarrows QUniFrm$
\item [[Quasi-uniformity]] in the point-free context via [[entourages]]

\end{enumerate}
\hypertarget{xiii_connectedness}{}\subsection*{{XIII. [[Connectedness]]}}\label{xiii_connectedness}

\begin{enumerate}%
\item A few observations about sublocales
\item [[Connected]] and disconnected locales
\item [[Locally connected]] locales
\item A weird example
\item A few notes

\end{enumerate}
\hypertarget{xiv_frame_of_reals_and_real_functions}{}\subsection*{{XIV. Frame of Reals and Real Functions}}\label{xiv_frame_of_reals_and_real_functions}

\begin{enumerate}%
\item The frame $L(R)$ of [[reals]]
\item Properties of $L(R)$
\item $L(R)$ versus the usual space of reals
\item The metric uniformity of $L(R)$
\item Continuous real functions
\item [[Cozero]] elements
\item More general real functions
\item Notes

\end{enumerate}
\hypertarget{xv_localic_groups}{}\subsection*{{XV. [[Localic Groups]]}}\label{xv_localic_groups}

\begin{enumerate}%
\item Basics
\item The category of [[localic groups]]
\item [[Closed Subgroup Theorem]]
\item The multiplication μ is open. The semigroup of open parts
\item Uniformities
\item Notes

\end{enumerate}
\hypertarget{appendix_i_posets}{}\subsection*{{Appendix I. [[Posets]]}}\label{appendix_i_posets}

\begin{enumerate}%
\item Basics
\item [[Zorn’s Lemma]]
\item [[Suprema]] and [[infima]]
\item [[Semilattices]], [[lattices]] and [[complete lattices]]. Completion
\item [[Galois connections]] (adjunctions)
\item (Semi)lattices as algebras. [[Distributive lattices]]
\item [[Pseudocomplements]] and [[complements]]. [[Heyting]] and [[Boolean algebras]]

\end{enumerate}
\hypertarget{appendix_ii_categories}{}\subsection*{{Appendix II. Categories}}\label{appendix_ii_categories}

\begin{enumerate}%
\item [[Categories]]
\item [[Functors]] and [[natural transformations]]
\item Some basic constructions
\item More special morphisms. Factorization
\item [[Limits]] and [[colimits]]
\item [[Adjunction]]
\item [[Adjointness]] and (co)limits
\item Reflective and [[coreflective subcategories]]
\item [[Monads]]
\item Algebras in a category

\end{enumerate}
category: reference



\end{document}