부산대학교 도서관

As the title promises, this book deals with a thorough study ofdifferent types of separation axioms in pointfree topology, i.e., therealm of the category ${\bf Frm}$ of frames and frame homomorphisms(which constitutes the algebraic side) and its opposite category, thecategory ${\bf Loc}$ of locales and localic maps (representing thetopological side of things). The present book can be seen as acontinuation of the monograph [{\it Frames and locales}, Front.Math., Birkhäuser/Springer Basel AG, Basel, 2012; MR2868166] by the same authors, although it is written in aself-contained way, providing the necessary background in the Appendix,which contains, among other things, a treatment of the construction ofthe binary coproduct in ${\bf Frm}$ as part of a more generalconstruction, exhibiting analogy with the tensor product.\par When working in ${\bf Loc}$, as in their previous book, the authorsuse their covariant treatment of locales, i.e., `with the arrowspointing the same way as in ${\bf Top}$', with a localic map being ameet-preserving map with a left Galois adjoint that preserves finitemeets (making localic maps exactly the right Galois adjoints of framehomomorphisms). Throughout the book, they choose to work either on thetopological (= localic) or algebraic (= frame) side of things,depending on which is more convenient for the topic at hand, which iselegant and pays off. An important role throughout the book is playedby the lattice of sublocales of a given locale, which happens to be acoframe (i.e., its dual is a frame, isomorphic to the lattice of framecongruences or the lattice of nuclei), which is one of the examples ofthe richness of the pointfree setting (even for spatial frames), andhere this covariant take on locales proves to be very useful, e.g.,when calculating in the sublocale lattice of a frame or when computingimages and pre-images of sublocales.\par The search for and study of separation axioms has, from the very startof locale theory, been a topic of great interest and activity to whichmany people, including the authors of this monograph, have madeimportant contributions. It showcases a lot of subtleties of pointfreetopology, exhibiting behavior that nicely extends that of classicaltopology (often with very different techniques and proofs being used)but also contains a lot of surprises or new things happening. Theauthors do an extremely good job collecting results scatteredthroughout the literature and presenting them here in a uniform way(for many of these results, it is the first time that they appear in aresearch monograph), drawing attention to the parallelisms anddifferences between the classical and pointfree worlds. The book startswith a very good sketch of the history of its subject matter providedin the Introduction, and also throughout the book, a lot of usefulhistorical comments and pointers to the literature are given. In orderto not overload this review, we will not mention the authors/papers towhich the concepts mentioned are due, since this is very welldocumented in the monograph under review.\par Chapter I (Separation in Spaces) deals with an overview of somewell-known facts of the classical separation axioms in ${\bf Top}$, butthe particular use of the $T_D$-axiom for the pointfree setting is alsohighlighted, since, for two $T_D$-spaces $X$ and $Y$, an isomorphism oftheir open set lattices $\Omega(X)$ and $\Omega(Y)$ implies that $X$and $Y$ are homeomorphic. Therefore, $T_D$ is the condition under whichsubspaces of a given $T_0$@-space $X$ are correctly represented assublocales of $\Omega(X)$. Throughout the book (unless explicitlystated otherwise), {\it topological spaces are always assumed to be$T_0$}, which is a very natural assumption when considering relationsto pointfree topology; we also use this convention here.\par Pointfree relatives of the classical lower separation axioms $T_1$ and$T_2$ (= Hausdorff) are discussed in detail in Chapter II (Subfitnessand Basics of Fitness), Chapter III (Axioms of Hausdorff Type) andChapter IV (Summarizing Low Separation).\par As the title of Chapter II suggests, the main ingredient is {\it subfitness} (sometimes also called {\it conjunctivity}), where a frame$L$ is called {\it subfit} if for every $a,b \in L$ with $a \nleq \,b$,there exists a $c \in L $ with $a \vee c = 1$ and $b \vee c \neq 1$.Subfitness for spatial frames turns out to be a mildly weaker relativeof $T_1$ since for a space $X$ being $T_1$ is equivalent to being $T_D$together with the subfitness of $\Omega(X)$. An importantcharacterization of subfitness, exploiting the rich structure of thesublocale lattice of a given locale, is treated in Section 4 of ChapterII, namely that a frame $L $ is subfit if and only if every opensublocale of $L$ is a join of closed sublocales. Fitness is alreadyintroduced in this chapter as well, but the authors convincingly arguethat fitness rather has to be considered as one of the higherseparation axioms, being akin to regularity (see Chapter VI). Fitnessand subfitness are related by, {\it inter alia}, the fact that a frameis fit if and only if all of its sublocales are subfit. Moreover,fitness is hereditary, whereas subfitness is not. It is also shown herethat fitness of a frame $L$ is equivalent to the condition that everyclosed sublocale of $L$ is a meet of open ones.\par Chapter III treats various approaches that have been taken to expressconcepts related to classical Hausdorffness (which we will call $T_2$for the rest of this paragraph) to the pointfree setting. After anintroductory paragraph which also mentions the rich history of thistopic, Section 2 treats three variants of weak Hausdorff properties,which turn out to be equivalent under subfitness, and which, takentogether with subfitness, are a conservative extension of $T_2$ (wherea property $\widehat{P}$ for frames is called a conservative extensionof a property $P$ for spaces, if a space $X$ has property $P$ if andonly if the frame $\Omega(X)$ has property $\widehat{P}$). Imitatingthe well-known characterization for $T_2$-spaces that the diagonal is aclosed subspace in the product of the space with itself yields thenotion of strong Hausdorffness for locales. As the choice ofterminology suggests, strong Hausdorffness of a frame impliesHausdorffness, and it has a lot of nice properties which are treated inSections 3, 9, 10 and 11 of this chapter, such as, e.g., the facts thatcompact strongly Hausdorff locales are regular, that compact sublocalesof strongly Hausdorff locales are closed, that dense framehomomorphisms are monomorphisms in the category of strongly Hausdorffframes and that strongly Hausdorff locales form an epireflectivesubcategory of ${\bf Loc}$. However, since for a space $X$ the frames$\Omega(X \times X) $ and $\Omega(X) \oplus \Omega(X)$ in general donot coincide, the notion of strong Hausdorffness fails to beconservative; in general, for a $T_0$-space $X$, only the implication`$\Omega(X)$ strongly Hausdorff $\Rightarrow X$ is $T_2$' holds. InSections 3, 6 and 7, a slightly weaker variant of strong Hausdorffnesswith very good behavior is discussed, which was originally obtained bydifferent authors looking at the topic from different angles, butresulted in equivalent concepts: one way builds on a spatial approachtreating $T_2$ as a form of weak regularity, and the other one replacesthe localic product and diagonal by a modification in the definition ofstrong Hausdorffness. The resulting notions are equivalent to thefollowing definition, calling a frame $L$ Hausdorff, if for any $1 \neqa \nleq b $ in $L$, there exists $u,v \in L$ with $u \nleq a $, $v\nleq b $ and $u \wedge v = 0$. This notion of Hausdorffness forlocales is a conservative extension of $T_2$, which is hereditary andproductive in ${\bf Loc}$; as a matter of fact, the Hausdorff localesform a reflective subcategory of ${\bf Loc}$. Chapter III also containsa short paragraph discussing the notion of point-Hausdorffness, whichis weaker than Hausdorffness (where a frame is called point-Hausdorffif every semi-prime element is maximal).\par Chapter IV very nicely rounds up and synthesizes this first part ofthe book on lower separation in the pointfree setting, providing, {\it inter alia}, some tables of the valid implications for easy reference.The higher separation axioms are the subject of the next four chapters:Chapter V (Regularity and Fitness), Chapter VI (Complete Regularity),Chapter VII (Normality) and Chapter VIII (More on Normality and RelatedProperties).\par Regularity, resp. complete regularity, of a frame $L$ is defined viathe rather below relation, resp. the completely below relation, in $L$by demanding that each $a \in L$ is the join of all elements ratherbelow $a$, resp. all elements completely below $a$. These provideconservative extensions of the classical regularity, resp. completeregularity, for spaces. Regularity implies strong Hausdorffness (andhence all the Hausdorff-like properties discussed in Chapter III andChapter IV) and in Chapter V, the authors dedicate two paragraphs toexplaining how several facts concerning density and compactness, whichcan be proved using weaker separation, become easier to deal with inthe presence of regularity. The authors then revisit the notion offitness, already treated in Chapter II, and show that it deserves to beconsidered as a relaxation of regularity. The chapter on regularity isconcluded by proving that the notions of fitness and regularity definereflective, resp. coreflective, subcategories of ${\bf Loc}$, resp.${\bf Frm}$. Chapter V discusses a (spatial) example showing thatcomplete regularity is properly stronger than regularity, as well as amethod for constructing a class of non-spatial completely regularframes, exhibiting that even in the presence of high separation, thedomain of applicability of the pointfree setting still considerablyexpands the classical one. Subsequent sections cover the equivalence ofcomplete regularity with uniformizability and the structure of thelattice of cozero elements of a frame $L$ (the pointfree counterpart tocozero sets in topology), which form a sub-$\sigma$-frame of $L$, andwhich generate $L$ if and only if $L$ is completely regular. Thechapter closes with three sections that really showcase some advantagesthat thinking pointfreely can bring about. The first one shows a moredesirable behavior on the pointfree side: the completely regularLindelöf frames (resp. locales) constitute a coreflective (resp.reflective) subcategory of the category of completely regularLindelöf frames (resp. locales). The second one indicates how,putting in extra work, localic thinking is a good setting for doingchoice-free (or even sometimes fully constructive) topology, which isillustrated in the last two sections of Chapter VI, dealing withchoice-free versions of complete regularity and compactification.\par Normality of a frame $L$ is defined in the obviously conservative wayas the condition that for all $a, b \in L$ with $a \vee b = 1$, thereare $u,v \in L$ such that $a \vee u = 1 = b \vee v$ and $u \wedge v =0$. In this case, subfitness does the trick of replacing $T_1$, sincenormal subfit frames are completely regular. In the remainder ofChapter VII, a pointfree version of the Wallman compactification andits relation to normality are treated, as well as the notion ofcomplete normality, which is shown to be equivalent to hereditarynormality.\par Chapter VII starts with sections on perfect normality andcollectionwise normality as relatives of normality; paracompactness isnot studied in this monograph since a chapter in the authors' earlierbook cited above is devoted to it. The frame of reals ${\cal L}({\BbbR})$ being discussed in detail in the Appendix, the authors now have attheir disposal the concept of a continuous real-valued function on aframe $L$, being a frame map ${\cal L}({\Bbb R}) \rightarrow L$, and thevery useful way of defining such functions via the technical concept of atrail. With ${\rm S}(L)$ denoting the sublocale lattice of $L$ (whichis a coframe), the notion of (arbitrary) real-valued function on aframe $L$ is defined as a continuous real-valued function on the frame${\rm S}(L)^{\rm op}$, i.e., a frame homomorphism ${\cal L}({\Bbb R})\rightarrow {\rm S}(L)^{\rm op}$, leading also to very naturaldefinitions of upper- and lower-semicontinuous real-valued functions on$L$ (such that continuous becomes equivalent to being upper and lowersemicontinuous). The next section of this chapter then develops apointfree version of the Katětov-Tong insertion theorem, statingthat normality of a frame $L$ is equivalent to the statement thatwhenever one considers an upper-semicontinuous real-valued function $f$on $L$ and a lower-semicontinuous real-valued function $g$ on $L$ with $f\leq g$, a continuous real-valued function $h$ on $L$ with $f \leq h\leq g$ can be found (here, an obvious identification of real-valuedcontinuous functions with certain real-valued functions is tacitlyassumed, as explained in detail in the monograph). Subsequently,pointfree versions of the characterizations of normality by Urysohn'sseparation lemma and Tietze's (bounded) extension theorem are nowreadily obtained as corollaries. The last two sections of this chapterdeal with extremal disconnectedness for frames (the definition of whichis obtained by formally dualizing the one of normality) and obtainingboth generalized and dualized versions of the Katětov-Tonginsertion theorem in this setting.\par The penultimate chapter is Chapter IX (Scatteredness: Joins of ClosedSublocales). In Chapter II, it is proved that subfitness of a locale isequivalent to the fact that every open sublocale of it is a join ofclosed ones. It is also proved that fitness can be expressed in asomewhat dual way, namely that a locale is fit if and only if every oneof its closed sublocales can be written as a meet of open ones.Surprisingly, and indicating that fitness is quite a lot stronger thansubfitness, it is proved that a locale is fit if and only if {\it every} one of its sublocales is a meet of open ones. This obviouslybegs the question of how the property that `every sublocale of $L$ canbe written as a join of closed ones' relates to this picture. Thisproperty turns out to be a very strong one, equivalent to $L$ beingscattered and fit, and also to $L$ being scattered and subfit (where aframe is called scattered if its sublocale lattice is a Booleanalgebra). In the remainder of the chapter, further interesting materialsuch as, e.g., the relation to classical scatteredness for spaces,Simmons' sublocale theorem and a study of the `Boolean cover' of asubfit frame is treated.\par In the last Chapter X (Subfit, Open and Complete), supplementaryresults are added, further completing the overall picture. As anexample, the Joyal-Tierney theorem is treated, which states that alocalic map $f\: M \rightarrow L$ is open (i.e., the image of everyopen sublocale of $M$ is open in $L$) if and only if its left Galoisadjoint $f^*\: L \rightarrow M$ is a complete Heyting homomorphism.Then building upon earlier material, one can infer that for $L$ subfit,a localic map $f\: M \rightarrow L$ is open if and only if its leftGalois adjoint $f^*\: L \rightarrow M$ is a complete latticehomomorphism.\par The book is very well written and pays great attention to detail,making it a pleasure to read and, on top of covering a vast body ofmaterial, it conveys a lot of insights through the many remarks,comments, cross-references and references to the literature appearingthroughout the book. Also, the ordering of the discussed topics is verywell thought through and the book contains an extensive bibliographyand a handy index plus a list of symbols.\par In summary, the monograph under review, together with its companion [op. cit.] can only be highly recommended; they constitute acomprehensive source of information and insights regarding pointfreetopology which will be of great interest and value to researchersalready working in the field, to mathematicians who want to studypointfree topology and to general topologists who want to see howpointfree topology relates to classical topology at the same time.