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We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$ . This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.

In this paper, we use Suzuki-type contraction to prove three fixed point theorems for generalized contractions in an ordered space equipped with two metrics; we obtain some generalizations of the Kannan fixed point theorem. Our results on partially ordered metric spaces generalize and extend some results of Ran and Reurings as well as of Nieto and Rodríguez-López. To illustrate the effectiveness of our main result, we give an application to matrix equations which involves monotone mappings.

[EN] Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps., The author would like to thank the referee for many helpful remarks and suggestions.

The main purpose of the present paper is to introduce new concept, “N-topological ordered spaces”. Also we define some of the separation axioms, weakly Nζ-T2-ordered space and N ζ-regularly ordered space in N-topological ordered spaces.

In this paper, we discuss the lexicographic ordered topologies on some products. We mainly prove: (1) The lexicographic ordered space λ γ is base-normal for any ordinals λ and γ. (2) The lexicographic ordered space [ 0 , 1 ) γ is Lindelof for each ordinal γ ≤ ω 1 . (3) The lexicographic ordered space λ γ is Lindelof for any λ ω 1 and γ ≤ ω 1 .

In our previous paper, in order to develop the pointfree theory of compactifications of ordered spaces, we introduced the concept of a proximity on a biframe as a generalization of the concept of a strong inclusion on a biframe. As a natural next step, we introduce the concept of a proximity morphism between proximity biframes. Like in the case of de Vries algebras and proximity frames, we show that the proximity biframes and proximity morphisms between them form a category PrBFrm in which composition is not function composition. We prove that the category KRBFrm of compact regular biframes and biframe homomorphisms is a proper full subcategory of PrBFrm that is equivalent to PrBFrm. We also show that PrBFrm is equivalent to the category PrFrm of proximity frames, and give a simple description of the concept of regularization using the language of proximity biframes. Finally, we describe the dual equivalence of PrBFrm and the category Nach of Nachbin spaces, which provides a direct way to construct compactifications of ordered spaces.

A proper location of the interval sections has significant impact on the traffic flow in the railway track network. This issue is critical during line modernization as well as when a new solution accounting for the traffic forecast at particular element of the railway track network is developed . However, the situation is more complex and more expensive for railway stations since improvement of the capacity requires critical organizational changes and additional investments. We have applied partially ordered linear spaces to study the capacity of a track system.

Urban ecosystems perform a distinctive metabolism appropriating fundamental materials and energy resources from other ecosystems. This appropriation enables a productive circularity which sustains the material production of urban space: urbanisation. Upon entering the urban ecosystem, the material fluxes are processed by human labour to be consumed, exchanged and accumulated. The appropriation of raw materials and the further production of technomass are entangled processes along a complex material circulation, sustained by internal metabolic processes of production, consumption, transformation and accumulation. The overall process couples several ecosystems along a metabolic urban network (MUN), a large-scale interconnected metabolism of urban and non-urban ecosystems. Long and short distance metabolic interactions configuring a huge space of flows from which distinct urbanisation patterns arise. The extended urbanisation, differential urbanisation, accumulative urbanisation, cascade urbanisation and speculative urbanisation are historically and ecologically determined. The MUN is an asymmetric spatiotemporal exchange structure to and from which matter, energy and information flow, producing profound socio-ecological asymmetries in a global range of dispersed urban tissues. The MUN is the vortex where the planetary processes of ecological and social deterioration is driven. The MUN is an analytical device to unveil urbanisation as a socio-ecological process of colonisation of urban and non-urban ecosystems.

We identify a class of subspaces of ordered spaces $\mathcal L$ for which the following statement holds: If $f:X\to L\in \mathcal L$ is a continuous bijections of a zero-dimensional space $X$, then $f$ can be re-routed via a zero-dimensional subspace of an ordered space that has weight not exceeding that of $L$.

We prove that countable i-weight reflects in continuous images of weight \(\le \omega _1\) for all Tychonoff spaces while separability reflects in continuous images of weight \(\le \omega _1\) for GO spaces. If X is a GO space and all continuous images of X of weight \(\le \kappa ^+\) have tightness at most \(\kappa \), then \(\mathrm{t}{(X)\le \kappa }\). All continuous images of weight \(\le \omega _1\) of a GO space X have countable pseudocharacter if and only if X is hereditarily Lindelof. Besides, all continuous images of weight \(\le \omega _1\) of a linearly ordered space X have \(G_\delta \)-diagonal if and only if X second countable.

We identify a condition on X that guarantees that any finite power of X is homeomorphic to a subspace of a linearly ordered space.

One of the central tasks in the theory of condensations is to describe topological properties that can be improved by condensation (i.e. a continuous one-to-one mapping). Most of the known counterexamples in the field deal with non-hereditary properties. We construct a countably compact linearly ordered (hence, monotonically normal, thus ” very strongly” hereditarily normal) topological space whose square and higher powers cannot be condensed onto a normal space. The constructed space is necessarily pseudocompact in all the powers, which complements a known result on condensations of non-pseudocompact spaces.

We study exchange economies in ordered normed spaces ( X , ‖ ⋅ ‖ ) where agents have possibly different consumption sets. We define the notion of semi-interior point of the positive cone X + of X , a notion weaker than the one of interior point and we study the existence of equilibrium in the case where X + has semi-interior points. In Section 4 , we study the case where X + has interior points and we prove a second welfare theorem and the existence of equilibrium. Subsequently we apply these results in the case where X + has semi-interior points. In the case of semi-interior points the supporting price vectors are continuous with respect to a new norm ∣ ∣ ∣ ⋅ ∣ ∣ ∣ on X which is strongly related with the initial norm and the ordering, and in some sense can be considered as an extension of the norm adopted in classical equilibrium models. Many examples of cones in normed and Banach spaces with semi-interior points but with empty interior are provided, showing that this class of cones is a rich one. In the last section we apply our results to strongly reflexive cones.

In this paper, we continue to study pairwise ($k$-semi-)stratifiable bitopological spaces. Some new characterizations of pairwise $k$-semi-stratifiable bitopological spaces are provided. Relationships between pairwise stratifiable and pairwise $k$-semi-stratifiable bitopological spaces are further investigated, and an open question recently posed by Li and Lin in \cite{LL} is completely solved. We also study the quasi-pseudo-metrizability of a topological ordered space $(X, \tau, \preccurlyeq)$. It is shown that if $(X, \tau, \preccurlyeq)$ is a ball transitive topological ordered $C$- and $I$-space such that $\tau$ is metrizable, then its associated bitopological space $(X,\tau^{\flat},\tau^{\natural})$ is quasi-pseudo-metrizable. This result provides a partial affirmative answer to a problem in \cite{KM}., Comment: 13 pages

We unify the concepts ofG-metric, metric-like, andb-metric to define new notion of generalizedb-metric-like space and discuss its topological and structural properties. In addition, certain fixed point theorems for two classes ofG-α-admissible contractive mappings in such spaces are obtained and some new fixed point results are derived in corresponding partially ordered space. Moreover, some examples and an application to the existence of a solution for the first-order periodic boundary value problem are provided here to illustrate the usability of the obtained results.

We identify a class of linearly ordered topological spaces X that may satisfy the property that X × X is homeomorphic to X ×l X or can be embedded into a linearly ordered space with the stated property. We justify the conjectures by partial results.

The main result of the paper says that if X is a paracompact GO-space, meaning a subspace of a linearly ordered space and M a paracompact space satisfying the first axiom of countability such that X can be embedded in M ω 1 then the product X × Y is paracompact for every paracompact space Y if and only if the first player of the G ( D C , X ) game, introduced by Telgarsky has a winning strategy. In particular we obtain that if X is paracompact GO-space of weight not greater than ω 1 then the product X × Y is paracompact for every paracompact space Y if and only if the first player of the G ( D C , X ) game has a winning strategy.

In this paper, we show that a generalized ordered space representable as the union of two closed monotonically Lindelöf subspaces is monotonically Lindelöf, which partially answers a question [7, Question 2] of Levy and Matveev. In addition, we show that the monotone Lindelöf property is hereditary with respect to open Lindelöf subsets in generalized ordered spaces.Keywords: Lindelöf, monotonically Lindelöf, generalized ordered spaces

En el interior de la Región de Murcia, aprovechando las condiciones hidrogeológicas de ciertos lugares (afloramientos del Trias o del Messiniense), en los que coincide material salino y una pequeña corriente de agua que lo atraviesa, se genera un medio de salmuera que tras su ordenación (salinas) permite las cosechas de sal. Su actividad y producciones han tenido importancia socioeconómica a nivel comarcal, y regional. En la segunda mitad del siglo XX, (mejoras de los transportes y mayor competencia de las salinas litorales) se produce el abandono de explotaciones salineras de interior como las de Sangonera la Seca (Murcia) y las de Librilla. De ellas apenas se conservan algunas infraestructuras del patrimonio material, como balsas o cocederos, y estanques o eras; que tras su rehabilitación serán testimonio de la transformación de espacios naturales en lugares culturales, con paisajes de interés educativo, ambiental, recreativo, turístico, etc. Inside the Region of Murcia, using the hydrogeological conditions of certain places (outcrop of the Trias or of the Messiniense), in which coincides salt material and a small stream that runs through it, a brine (salt) environment is generated that after its ordination, allows the salt harvest. Activities related of the salt indoor spaces have played an important social-economic role at regional levels. In the second half of the twentieth century, (due to transport improvement and more competitive litoral salt harvests), the abandonment of inland salt mines as the ones of Sangonera (Murcia) and Librilla occurred. Only a few of the heritage infrastructures related to these inland salt mines such as rafts or ponds are still preserved; after his rehabilitation will be witness of the transformation from natural spaces to cultural places, with landscapes of educational, environmental, recreational, tourist attractions, etc.

It is shown that a mixed monotone property in coupled fixed point results can be replaced by another property which is automatically satisfied in the case of a totally ordered space, the case which is the most important in applications. Hence, these results can be applied in a much wider class of problems.

This paper contributes to the algebraization of topology via the theory of monads and lax extensions of monads and their associated lax algebras (see Barr (1970) [1] , Clementino and Hofmann (2003) [2] , Clementino, Hofmann and Tholen (2004) [4] , Clementino and Tholen (2003) [5] , Lowen and Vroegrijk (2008) [11] , Manes (1974) [12] , Seal (2005) [14] ). We construct a monad P , a lax extension P ¯ and monad morphisms into P from the most important monads as studied in the aforementioned papers such that their lax extensions and their associated categories of lax algebras can be derived from the extension P ¯ by initial lifts via these monad morphisms. This provides us with a completely unified way to obtain the categories Top , App , Met and Ord without the necessity to leave the realm of Rel as was previously required in Clementino and Hofmann (2003) [2] , Clementino, Hofmann and Tholen (2004) [4] and Clementino and Tholen (2003) [5] in particular in order to obtain App and Met .

For any generalized ordered space X with the underlying linearly ordered topological space Xu, let X* be the minimal closed linearly ordered extension of X and $\tilde {X}$ be the minimal dense linearly ordered extension of X. The following results are obtained. (1)The projection mapping π:X*→X, π(〈x,i〉)=x, is closed.(2)The projection mapping $\phi : \tilde {X} \rightarrow X_u$, ϕ(〈x,i〉)=x, is closed.(3)X* is a monotone D-space if and only if X is a monotone D-space.(4)$\tilde {X}$ is a monotone D-space if and only if Xu is a monotone D-space.(5)For the Michael line M, $\tilde {M}$ is a paracompact p-space, but not continuously Urysohn.

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56–65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorcuk (Soviet Math Dokl 7:1011–1014, 1966; Sib Math J 10:124–132, 1969) and Kaufman (Colloq Math 17:35–39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13:56–65, 1975) and Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.

The main goal of this article is to reveal the significance of height and verticality history of skyscrapers and skylines. The objectives are as follows: 1. trace the origin of design concepts related to skyscraper; 2. discuss the perceived experience of the cultural aspects of skyscrapers and skylines; 3. describe the differences and similarities of the profiles of with comparable skylines. The methodology of study is designed to explore the perceived theory and principals of skyscraper and skyline development phenomenon and its key features. The skyscraper reveals an assertive creative form of vertical design. Skyscraper construction also relates to the origin of ancient cultural symbolism as the dominant vertical element as the main features of an ordered space. The historical idea of height reaches back to the earliest civilization such as the Tower of Babel. Philosophical approaches of elements of such post-structuralism have been included in studying of skyscraper phenomenon. The analysis of skyscraper and their resulting skyline are examined to show the connection to their origins with their concepts of height and verticality. From the historical perspective, cities with skyscrapers and a skyline turn out to be an assertive manifestation of common ideas of height and verticality.

We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace., Comment: We have rewritten and substantially expanded the manuscript

We propose a pair of axioms (O.p.1) and (O.p.2) for 1 ≤ p ≤ ∞ and initiate a study of a (matrix) ordered space with a (matrix) norm, in which the (matrix) norm is related to the (matrix) order. We call such a space a (matricially) order smooth p-normed space. The advantage of studying these spaces over Lp-matricially Riesz normed spaces is that every matricially order smooth ∞-normed space can be order embedded in some C*-algebra. We also study the adjoining of an order unit to a (matricially) order smooth ∞-normed space. As a consequence, we sharpen Arveson’s extension theorem of completely positive maps. Another combination of these axioms yields an order theoretic characterization of the set of real numbers amongst ordered normed linear spaces.

The 2002 government restoration of Suq in Damascus led to disagreements between officials and merchants over acceptable spatial practice in the marketplace. Though both were vested in the Suq's introduction to the historic Old City, they had different interpretations of this role. Officials emphasized a sanitized and ordered space whereas merchants focused on the commercial activities necessary in the marketplace over discipline. Eventually both interpretations, though at times contradictory, were included in the project. This article illustrates how the replication of modernity in a public space with inherent contradictions is a result of negotiations between formal and informal modernitites.

espanolEl siguiente analisis de la narracion del Reino Medio conocida como Historia de Sinuhe presenta una lectura en la que se proponen algunas consideraciones sobre el texto como un marco de referencia espacial. Estas consideraciones se basan en las afinidades de texto y territorio en cuanto que ambos son habitados por el individuo que los usa como escenario en el cual sus cambios fisicos espaciales y temporales tienen lugar. El movimiento en un espacio fisico puede compararse al movimiento de la mano y del instrumento de la escritura sobre el soporte escriturario gobernados ambos por la mano del autor. Una caracteristica del espacio ordenado es la existencia en su seno de hitos y que lo dividen y que separan diferentes areas geograficas. En el analisis que sigue de la Historia de Sinuhe se propone la existencia de tales hitos en el seno del texto. Las areas creadas y separadas por estos hitos pueden caracterizarse por las acciones que el protagonista lleva a cabo en ellas, mientras que lo opuesto tambien tiene lugar, es decir, la presencia del protagonista caracteriza esas zonas por medio de su evolucion y situacion en ellas. Estos aspectos en su conjunto permiten establecer un mapa de la estructura narrativa del texto, resultando asi en un conjunto de zonas y de hitos que se caracterizan etica y moralmente de modo mutuo dando como resultado la relacion metaforica entre texto y territorio que aqui se propone. EnglishThe following analysis of the Middle Kingdom narrative known as the Tale of Sinuhe presents a reading in which some reflections on the text as spatial frame are proposed. These considerations are based on the affinity of text and territory in so far as both are inhabited by the individual who uses them as a scenario within which his physical changes in space and time take place. Movement in a physical space can be compared with the motion of the hand and the writing tool on the writing medium, both governed by the mind of the author. One characteristic of ordered space is the existence within it of landmarks dividing and separating different geographic areas. The following analysis of the Tale of Sinuhe suggests the existence of these landmarks within the text. The areas created and separated by these landmarks can be characterized by the actions of the individual (Sinuhe) performed within them; the reverse also takes place, in that the individual is characterized by particular aspects of the area inhabited at a given moment in the course of the narration. These aspects, taken in conjunction, allow for the mapping of the narrative structure of the text as an interpretational device, resulting in a conjunction of areas and landmarks which characterize the individual and are characterized by him, thus producing the metaphorical relationship between text and territory here proposed.