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This paper considers the compact hedgehog as a frame presented by generators and relations, based on the presentation of the frame of extended real numbers. The main focus will be on the point-free version of continuous and semicontinuous functions that arise from it, and their application in characterizations of variants of collectionwise normality. The variants to be considered are defined by selections of adequate families of sublocales and their characterizations depend on lattice-theoretic properties of the selected families. This way insertion and extension results for semicontinuous and continuous functions with values in the compact hedgehog frame are generalized and unified.

espanolGecarcinus quadratus es un cangrejo terrestre depredado por algunas especies, sin embargo, hay poca informacion en el nivel de genero y especie de sus predadores. Esta nota cientifica presenta el primer registro de Aramides cajaneus depredando a G. quadratus. La observacion ocurrio durante el dia, en mayo del 2021, en Costa Rica y contribuye al conocimiento sobre la historia natural, comportamiento y la ecologia de estas especies. portuguesO Gecarcinus quadratus e um caranguejo terrestre predado por algumas especies, no entanto, ha pouca informacao sobre o genero e o nivel de especies de seus predadores. Esta nota cientifica apresenta o primeiro registro de Aramides cajaneus predando o G. quadratus. A observacao ocorreu durante o dia, em maio de 2021, na Costa Rica e contribui para o conhecimento sobre a historia natural, o comportamento e a ecologia dessas especies. EnglishGecarcinus quadratus is a land crab preyed upon by some species, but there is little information about the genera and species of these predators. This scientific note presents the first record of Aramides cajaneus preying upon G. quadratus. This observation occurred during daylight in May 2021 in Costa Rica, and contributes to the knowledge about the natural history, behavior and ecology of these species.

IRET The LIFE Index can be useful to determine the effect of changes in river flow on the community of aquatic macroinvertebrates and as a tool for the implementation of environmental flows. This study demonstrates how to classify aquatic macroinvertebrates into a flow category in order to adjust the LIFE Index to Costa Rica. A panel of experts was surveyed to classify the most common genera into a water velocity category, based on their experience. Also, for one-year, aquatic macroinvertebrates were collected in the low and middle basin of the Naranjo River under different velocities, and by using the TITAN2 package, their respective thresholds for current velocity were determined. Variation was observed in the responses of the expert panel; however, several taxa overlapped by more than 60% between the expert classification and the TITAN2 test results. The TITAN2 test assigned a velocity threshold to 32 genera, with the inflection point being 0.1 m/s. The expert panel served as a tool to assign a category to those genera that the TITAN2 test did not contemplate. Organisms without morphological adaptations to survive fast flowing conditions decreased in frequency above 0.1 m/s, while genera related to moderate and high velocities increased in frequency. Through the panel of experts and the TITAN2 test, it was possible to assign the most common genera in the country to a current velocity category and therefore adjust the LIFE index. El Índice LIFE puede ser útil para determinar el efecto de los cambios en el caudal de los ríos en la comunidad de macroinvertebrados acuáticos y como herramienta para la implementación de caudales ambientales. Este estudio demuestra cómo clasificar a los macroinvertebrados acuáticos en una categoría de flujo para ajustar el Índice LIFE a Costa Rica. Se encuestó a un panel de expertos para clasificar los géneros más comunes en una categoría de velocidad del agua, según su experiencia. Asimismo, durante un año, se recolectaron macroinvertebrados acuáticos en la cuenca baja y media del río Naranjo a diferentes velocidades, y mediante el uso del paquete TITAN2 se determinaron sus respectivos umbrales de velocidad de corriente. Se observó variación en las respuestas del panel de expertos; sin embargo, varios taxones se superpusieron en más del 60% entre la clasificación de expertos y los resultados de la prueba TITAN2. La prueba TITAN2 asignó un umbral de velocidad a 32 géneros, con un punto de inflexión de 0,1 m / s. El panel de expertos sirvió como herramienta para asignar una categoría a aquellos géneros que la prueba TITAN2 no contemplaba. Los organismos sin adaptaciones morfológicas para sobrevivir en condiciones de flujo rápido disminuyeron en frecuencia por encima de 0,1 m / s, mientras que los géneros relacionados con velocidades moderadas y altas aumentaron en frecuencia. A través del panel de expertos y la prueba TITAN2, fue posible asignar los géneros más comunes en el país a una categoría de velocidad actual y así ajustar el índice LIFE. Universidad Nacional, Costa Rica Universidad de Costa Rica, Costa Rica Instituto Regional de Estudios en Sustancias Tóxicas

The fact that equalizers in the context of strongly Hausdorff locales (similarly like those in classical spaces) are closed is a special case of a standard categorical fact connecting diagonals with general equalizers. In this paper we analyze this and related phenomena in the category of locales. Here the mechanism of pullbacks connecting equalizers is based on natural preimages that preserve a number of properties (closedness, openness, fittedness, complementedness, etc.). Also, we have a new simple and transparent formula for equalizers in this category providing very easy proofs for some facts (including the general behavior of diagonals). In particular we discuss some aspects of the closed case (strong Hausdorff property), and the open and clopen one.

We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting.

The hedgehog metric topology is presented here in a pointfree form, by specifying its generators and relations. This allows us to deal with the pointfree version of continuous (metric) hedgehog-valued functions that arises from it. We prove that the countable coproduct of the metric hedgehog frame with κ spines is universal in the class of metric frames of weight κ ⋅ ℵ 0 . We then study κ -collectionwise normality, a cardinal extension of normality, in frames. We prove that this is the necessary and sufficient condition under which Urysohn separation and Tietze extension-type results hold for continuous hedgehog-valued functions. We show furthermore that κ-collectionwise normality is hereditary with respect to F σ -sublocales and invariant under closed maps.

The property to be discussed in this chapter is very restrictive, so restrictive that one may hesitate to think about it as of an honest separation axiom (similarly like one does not usually think of discreteness as of a separation property: it says that distinct points can be separated by closed sets, but this is really somehow going too far).

We can only agree with Peter Johnstone who wrote in Johnstone (Bull Amer Math Soc (N.S.) 8:41–53, 1983) that

Here we start with two more variants of normality. There is the perfect normality, which turns out to be a conjunction of the classical perfectness (which is slightly different in the point-free context due to the different behaviour of sublocales and subspaces) and normality; in a way it can be viewed as a weaker form of metrizability. Next we deal with the technically important collectionwise normality. Then, in the penultimate section we prove and discuss the Katětov–Tong insertion theorem, using (to advantage) the techniques of the point-free real line. We finish with a certain duality between normality and extremal disconnectedness that allows to translate several results concerning normality to facts about extremal disconnected frames.

So far (with just a few exceptions) we have discussed the “low separation axioms”: those that can be used to replace the classical T1, and several conditions mimicking the Hausdorff property. Before turning to the stronger ones we will now summarize relations between them and add some facts that will help to understand the role they play and can play in further applications.

Point-free topology is the study of the category of locales and localic maps and its dual category of frames and frame homomorphisms. These notes cover the topics presented by the first author in his course on Frames and Locales at the Summer School in Algebra and Topology. We give an overview of the basic ideas and motivation for point-free topology, explaining the similarities and dissimilarities with the classical setting and stressing some of the new features.

In this final chapter, after briefly summarizing some of the already discussed facts concerning subfitness and fitness, we will tackle an aspect of these properties we have not examined yet, the role they play in the links of the phenomena of completeness, openness and the Heyting structure. Let us explain the main topic we will be interested in.

We will start by recalling some standard separation axioms in topological spaces X and discussing how to cope with them without points. In the point-free setting we think of classical spaces in terms of the complete lattices Ω(X) of their open sets, and of general spaces as of complete lattices of similar nature. Hence, the fact that the classical separation is expressed by statements in which individual points (and non-open subsets) play a prominent role seems to be a principal obstacle for extending them to the more general context. But the situation is much more favourable than what one may expect. First of all, some of the separation conditions can be replaced by obviously equivalent statements using the calculus of the lattice only: such is an absolutely straightforward reformulation of normality, very easy (and classically transparent) reformulation of regularity, and a translation of complete regularity which needs some more explanation (but this explanation concerns the role of real functions which calls for explanation in the classical situation as well). This will be presented already in this chapter. Then there are reformulations and replacements that are more involved (in particular the Hausdorff axiom that will come in variants with nontrivial relations). And then there are specific point-free separation conditions that are of particular interest: some of them akin to classical ones but not quite equivalent, some of them naturally arising from lattice theoretic requirements. They will be discussed and analysed in the individual chapters below (to be more exact: regularity, complete regularity and normality will also have their individual chapters; what we have in mind is that their reformulations can be presented right away while the others will be postponed).

Among the conditions of separation type, the axiom of regularity is very special. From the earliest stages of point-free topology there was natural interest in conditions that would capture classical separation phenomena as convincingly as possible in the new, more general, context. Classical separation axioms are typically formulated in the language of points and point-dependent notions; hence, one looked for equivalent formulations, or imitated the geometric intuition to obtain suitable replacements or at least analogies (see, e.g., Isbell (Math Scand 31:5–32, 1972), Dowker and Strauss. Separation axioms for frames. In: Topics in Topology, pp. 223–240. Proc. Colloq., Keszthely, 1972. Colloq. Math. Soc. Janos Bolyai, vol. 8, North-Holland, Amsterdam, 1974, Isbell (Math Scand 36:317–339, 1975), Simmons. A framework for topology. In: Logic Colloq. ’77, pp. 239–251. Stud. Logic Foundations Math., vol. 96. North-Holland, Amsterdam-New York, 1978, Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge 1982, Rosický and Smarda (Math. Proc. Cambridge Philos. Soc. 98:81–86, 1985)). In this company, regularity stands out. As we have already seen in Chap. I, it can be translated very easily, and the obtained formula has a natural appeal even in classical spaces (in fact, in an obviously equivalent form it is used classically anyway). There is no reasonable doubt that this formula makes a fully satisfactory point-free extension. It can be used without problems for proving useful facts parallel with the classical ones; moreover, it is algebraically versatile and easy to work with.

Exact meets in a distributive lattice are the meets ⋀ i a i such that for all b, ( ⋀ i a i ) ∨ b = ⋀ i ( a i ∨ b ) ; strongly exact meets in a frame are preserved by all frame homomorphisms. Finite meets are, trivially, (strongly) exact; this naturally leads to the concepts of exact resp. strongly exact filters closed under all exact resp. strongly exact meets. In [2] , [12] it was shown that the subsets of all exact resp. strongly exact filters are sublocales of the frame of up-sets on a frame, hence frames themselves, and, somewhat surprisingly, that they are isomorphic to the useful frame S c ( L ) of sublocales join-generated by closed sublocales resp. the dual of the coframe meet-generated by open sublocales. In this paper we show that these are special instances of much more general facts. The latter concerns the free extension of join-semilattices to coframes; each coframe homomorphism lifting a general join-homomorphism φ : S → C (where S is a join-semilattice and C a coframe) and the associated (adjoint) colocalic maps create a frame of generalized strongly exact filters (φ-precise filters) and a closure operator on C (and – a minor point – any closure operator on C is thus obtained). The former case is slightly more involved: here we have an extension of the concept of exactness (ψ-exactness) connected with the lifts of ψ : S → C with complemented values in more general distributive complete lattices C creating, again, frames of ψ-exact filters; as an application we learn that if such a C is join-generated (resp. meet-generated) by its complemented elements then it is a frame (resp. coframe) explaining, e.g., the coframe character of the lattice of sublocales, and the (seemingly paradoxical) embedding of the frame S c ( L ) into it.

The purpose of this paper is to identify the role of perfectness in the Michael insertion theorem for perfectly normal locales. We attain it by characterizing perfect locales in terms of strict insertion of two comparable lower semicontinuous and upper semicontinuous localic real functions. That characterization, when combined with the insertion theorem for normal locales, provides an improved formulation of the aforementioned pointfree form of Michael’s insertion theorem.

Introducción: La distribución de los macroinver tebrados acuáticos en los ríos puede estar determinada por el tipo del sustrato, la forma del canal del río, el régimen hidrológico y las interacciones con factores bióticos y abióticos. En Costa Rica se desconoce en gran parte de los ríos, las variaciones en la distribución de los macroinvertebrados acuáticos producto de las transiciones y consecuentes variaciones en características fisicoquí micas e hidrológicas. Objetivos: 1) Estudiar el cambio del ensamblaje de macroinvertebrados acuáticos en dos sitios dentro del cauce del Río Naranjo, durante los cuatro periodos hidrológicos del año; 2) determinar la relación entre las variables fisicoquímicas, sustrato y el caudal con el ensamblaje. Metodología: El estudio se realizó en el Río Naranjo, Costa Rica, en dos sitios de muestreo durante un año, evaluando los cuatro periodos hidrológicos del año. En cada sitio de muestreo se obtuvieron valores de veloci dad y profundidad del agua, caudal, oxígeno disuelto, pH, conductividad, sulfatos, amonio, nitrato, silicatos, fosfatos y sustrato, y se obtuvo una muestra de macroinvertebra dos acuáticos por cada tipo de microhábitat determinado por la velocidad del agua. Se realizó una prueba PER MANOVA de dos vías, para determinar si existe una diferencia entre la abundancia de macroinvertebrados acuáticos, por periodo y entre sitios. Se hizo un análisis de correspondencia canónica para observar la relación entre los macroinvertebrados acuáticos y los factores ambien tales. Por último, se realizó una prueba de Spearman para determinar la relación positiva o negativa de las variables con los macroinvertebrados Resultados: Hubo una mayor abundancia durante la estación seca, mientras que durante la transición de la estación seca a la lluviosa se registró la menor abundancia de organismos. El ACC indicó que el primer eje presentó una relación positiva con el sustrato limo y arena, y una relación negativa para las variables velocidad de corriente y sustrato rocoso. El segundo eje presentó una relación positiva con el caudal y los nitratos, y negativa para el sustrato canto rodado. La mayoría de los organismos mostraron una respuesta negativa al aumento del flujo y los nitratos. Discusión: Conforme aumenta el caudal, aumenta la velocidad del agua y no todos los orga nismos registrados en el Río Naranjo cuentan con carac terísticas para sobrevivir a estas condiciones. Además, la profundidad también aumenta junto con la fuerza de arras tre, provocando el movimiento de los sustratos de menor tamaño y una remoción de organismos. El aumento de nitratos puede generar una disminución de los organismos más sensibles como Plecoptera y Ephemeroptera. A pesar de las variaciones fisicoquímicas y de caudal registradas en el Río Naranjo, éstas no fueron lo suficientemente fuertes como para variar el ensamblaje de macroinvertebrados acuáticos, por lo que la cuenca por el momento no cuenta con factores estresantes que elimine organismos durante uno o varios meses del año. Introduction: The distribution of aquatic macroinvertebrates in rivers may be determined by the type of substrate, the shape of the river channel, the hydrological regime, and interactions with biotic and abiotic factors. Variations in the distribution of aquatic macroinvertebrates due to transitions and variations in physicochemical and hydrological characteristics are unknown in most rivers. Objectives: 1) to study the change in the aquatic macroinvertebrates assemblages in two sites in Naranjo River, during the four hydrological seasons; 2) to determine the relationship between the physicochemical variables, substrate and flow with the assemblages. Methodology: The study was conducted in the Naranjo River, Costa Rica, at two sampling sites during one year, evaluating the four seasons of the year. In each station we measure, values of water velocity and depth, flow rate, dissolved oxygen, pH, conductivity, sulfates, ammonium, nitrate, silicates, phosphates, and substrate were obtained, and a sample of aquatic macroinvertebrates was obtained for each type of microhabitat determined by water velocity. Two-way PERMANOVA was made to determine if there is a difference among aquatic macroinvertebrate abundance, per period and between sites. Canonical correspondence analysis was used to observe the relationship between aquatic macroinvertebrates and environmental factors. Finally, a Spearman’s rank correlation coefficient was performed to determine the positive or negative relationship of variables with macroinvertebrates. Results: There was a greater abundance during the dry season, while during the transition from dry to the rainy season, the lowest abundance of organisms was recorded. The CCA indicates that the first axis presented a positive relationship with the silt and sand substrate and a negative relationship for the current velocity and rock substrate variables. The second axis showed a positive relationship with flow and nitrates and negative for the pebble substrate. Most organisms showed a negative response to increased flow and nitrates. Discussion: As flow increases, current velocity increases, and not all organisms recorded in the Naranjo River have the characteristics to survive these conditions. In addition, the depth also increases along with the drag force, causing movement of smaller substrates and removal of organisms. Increased nitrates can lead to a decrease in more sensitive organisms, such as Plecoptera and Ephemeroptera. In spite of the physicochemical and flow variations registered in the Naranjo River, these were not strong enough to vary the assemblages of aquatic macroinvertebrates, so the basin does not have stressful factors that eliminate organisms during one or several months of the year. UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Biología UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigación en Ciencias del Mar y Limnología (CIMAR)

Hereditary extremal disconnectedness of frames and its equivalent form of complete extremal disconnectedness are the topic of this paper. We study them in parallel with the corresponding normality properties of frames. Among several characterizations, we show that a frame is hereditarily normal [resp., hereditarily extremally disconnected] if and only if all its open and dense sublocales [resp., closed and dense sublocales] are normal [resp., extremally disconnected]. Even if spaces typically have more sublocales than subspaces, hereditary normality and hereditary extremal disconnectedness are shown to be conservative extensions of the classical properties. Furthermore, we provide such a general setting that permits us to treat several variants of the concepts under study in a unified way. Some of the presented results are new for the traditional topological spaces.

We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one).

Remainders of subspaces are important e.g. in the realm of compactifications. Their extension to pointfree topology faces a difficulty: sublocale lattices are more complicated than their topological counterparts (complete atomic Boolean algebras). Nevertheless, the co-Heyting structure of sublocale lattices is enough to provide a counterpart to subspace remainders: the sublocale supplements. In this paper we give an account of their fundamental properties, emphasizing their similarities and differences with classical remainders, and provide several examples and applications to illustrate their scope. In particular, we study their behavior under image and preimage maps, as well as their preservation by pointfree continuous maps (i.e. localic maps). We then use them to characterize nearly realcompact and nearly pseudocompact frames. In addition, we introduce and study hyper-real localic maps.

This paper extends the extension theorem for localic real functions of J. Gutierrez Garcia and T. Kubiak [8] from complemented sublocales to arbitrary sublocales. As an application, the theory of point-free C-, C ⁎ -, and z-embeddings is revisited.

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform sublocale can be extended to a (regular) Cauchy point on the larger (quasi-)uniform frame.

The frame Sc(L) generated by closed sublocales of a locale L is known to be a natural Boolean ("discrete") extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and Sc(L) are isomorphic. The construction Sc is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms Sc(L) → Sc(M). This is trivial for Boolean L and easy for a wide class of spatial L, M. Then, we show that one can lift all h : L → 2 for weakly Hausdorff L (and hence the spectra of L and Sc(L) are naturally isomorphic), and finally present liftings of h : L → M for regular L and arbitrary Boolean M.Mathematics Subject Classification (2010): 06D22, 54D10, 54D35.Keywords: Frame, locale, sublocale, sublocale lattice, essential extension, subfit, Booleanization

We provide the appropriate unifying framework for the various descriptions of the Dedekind completion of the ring C( L ) of continuous real functions on a frame L . It is based on suitable Galois connections and a general result about Galois connections, showing once more the ubiquity of (Galois) adjunctions between partially ordered sets and their conceptual simplicity and effectiveness. Mathematics Subject Classification (2010) : Primary: 06D22; Secondary: 26A15, 54C30, 54D15. Keywords: Frame, locale, frame of reals, continuous real function, function ring, order complete, Dedekind completion, scale, normal semicontinuous real function, partial real function, Hausdorff continuous real function, cb-frame.

This paper contains some new facts about subfitness and weak subfitness. In the case of spaces, subfitness is compared with the axiom of symmetry, and certain seeming discrepancies are explained. Further, Isbell’s spatiality theorem in fact concerns a stronger form of spatiality (T 1-spatiality) which is compared with the T D -spatiality. Then, a frame is shown to be subfit iff it contains no non-trivial replete sublocale, and the relation of repleteness and subfitness is also discussed in spaces. Another necessary and sufficient condition for subfitness presented is the validity of the meet formula for the Heyting operation, which was so far known only under much stronger conditions.

It is well known that a locale is subfit iff each of its open sublocales is a join of closed ones, and fit iff each of its closed sublocales is a meet of open ones. This formulation, however, exaggerates the parallelism between the behavior of fitness and subfitness. For it can be shown that a locale is fit iff each of its sublocales is a meet of closed ones, but it is not the case that a locale is subfit iff each of its sublocales is a join of closed ones. Thus we are led to take up the very natural question of which locales have the feature that every sublocale is a join of closed sublocales. In this note we show that these are precisely the subfit locales which are scattered in the pointfree sense of (13), and we add a variation for spatial frames.

Several familiar results about normal and extremally disconnected (classical or pointfree) spacesshape the idea that the two notions are somehow dual to each other and can therefore be studied inparallel. In this talk we discuss the source of this ‘duality’ and show that each pair of parallelresults can be framed by the ‘same’ proof. The key tools for this purpose are relative notionsof normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixedclass of complemented sublocales of the given locale) that bring and extend to locale theory avariety of well-known classical variants of normality and upper and lower semicontinuities in ailluminating unified manner. This approach allows us to unify under a single localic proof a great variety ofclassical insertion results, as well as their corresponding extension results.

Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or $$T_1$$ . Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting.

Products of locales (generalized spaces) are coproducts of frames. Because of the algebraic nature of the latter they are often viewed as algebraic objects without much topological connotation. In this paper we first analyze the frame construction emphasizing its tensor product carrier. Then we show how it can be viewed topologically, that is, in the sum-of-the-open-rectangles perspective. The main aim is to present the product from different points of view, as an algebraic and a geometric object, and persuade the reader that both of them are fairly transparent.

This paper makes a comparison between two notions of perfectness for locales which come as direct reformulations of the two equivalent topological definitions of perfectness. These reformulations are no longer equivalent. It will be documented that a locale may appropriately be called perfect if each of its open sublocales is a join of countably many closed sublocales. Certain circumstances are exhibited in which both reformulations coincide. This paper also studies perfectness in mildly normal locales. It is shown that perfect and mildly normal locales coincide with the Oz locales extensively studied in the last decade.Mathematics Subject Classication (2010): 06D22, 54D15.Key words: Locale, sublocale, Fα-sublocale, Gα-sublocale, normality, mild normality,perfectness, perfect normality, pm-normality, Oz locale.

Generalizing the obvious representation of a subspace $${Y \subseteq X}$$ as a sublocale in Ω(X) by the congruence $${\{(U, V ) | U\cap Y = V \cap Y\}}$$ , one obtains the congruence $${\{(a, b) |\mathfrak{o}(a) \cap S = \mathfrak{o}(b) \cap S\}}$$ , first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets $${S\subseteq L}$$ . The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective.

Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore be studied in parallel. This paper investigates the source of this ‘duality’ and shows that each pair of parallel results can be framed by the ‘same’ proof. The key tools for this purpose are relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed class of complemented sublocales of the given locale) that bring and extend to locale theory a variety of well-known classical variants of normality and upper and lower semicontinuities in an illuminating unified manner. This approach allows us to unify under a single localic proof all classical insertion, as well as their corresponding extension results.

Monotone normality is usually defined in the class of T 1 spaces. In this paper we study it under the weaker condition of subfitness, a separation condition that originates in pointfree topology. In particular, we extend some well known characterizations of these spaces to the subfit context (notably, their hereditary property and the preservation under surjective continuous closed maps) and present a similar study for stratifiable spaces, an important subclass of monotonically normal spaces. In the second part of the paper, we extend further these ideas to the lattice theoretic setting. In particular, we give the pointfree analogues of the previous results on monotonically normal spaces and introduce and investigate the natural pointfree counterpart of stratifiable spaces.

The concepts of fitness and subfitness (as defined in Isbell, Trans. Amer. Math. Soc. 327, 353–371, 1991) are useful separation properties in point-free topology. The categorical behaviour of subfitness is bad and fitness is the closest modification that behaves well. The separation power of the two, however, differs very substantially and subfitness is transparent and turns out to be useful in its own right. Sort of supplementing the article (Simmons, Appl. Categ. Struct. 14, 1–34, 2006) we present several facts on these concepts and their relation. First the “supportive” role subfitness plays when added to other properties is emphasized. In particular we prove that the numerous Dowker-Strauss type Hausdorff axioms become one for subfit frames. The aspects of fitness as a hereditary subfitness are analyzed, and a simple proof of coreflectivity of fitness is presented. Further, another property, prefitness, is shown to also produce fitness by heredity, in this case in a way usable for classical spaces, which results in a transparent characteristics of fit spaces. Finally, the properties are proved to be independent.