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Given an arbitrary spectral space $X$, we endow it with its specialization order $\leq$ and we study the interplay between suprema of subsets of $(X,\leq)$ and the constructible topology. More precisely, we investigate about when the supremum of a set $Y\subseteq X$ exists and belongs to the constructible closure of $Y$. We apply such results to algebraic lattices of sets and to closure operations on them, proving density properties of some distinguished spaces of rings and ideals. Furthermore, we provide topological characterizations of some class of domains in terms of topological properties of their ideals.

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian {\ell}-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countable distributive lattice D with zero is isomorphic to the lattice of all principal ideals of an Abelian {\ell}-group (we say that D is {\ell}-representable) iff for all a, b $\in$ D there are x, y $\in$ D such that a $\lor$ b = a $\lor$ y = b $\lor$ x and x $\land$ y = 0. On the other hand, we construct a non-{\ell}-representable bounded distributive lattice, of cardinality $\aleph$ 1 , with an {\ell}-representable countable L$\infty, \omega$-elementary sublattice. In particular, there is no characterization, of the class of all {\ell}-representable distributive lattices, in arbitrary cardinality, by any class of L$\infty, \omega$ sentences., Comment: Misprints v2: In Example 7.1, (a-mb)\wedge(b-mc) \leq 0 (i.e., \wedge instead of \vee).In Corollary 8.6, X, Y^-, and Y^+ are just elements of \Op(\mathcal{H}) (not necessarily basic open)

The topological embedding of a topological semigroup S, commutative with the property of cancelation, into the group G = S × S / R , (R the equivalence ( a , b ) R ( a ′ , b ′ ) ⇔ ab ′ = a ′ b ) to which S is algebraically embedded, was the subject of the search for the mathematicians of a long period. It was based on the fact that S must naturally be a uniform topological space, as every topological group was. The present paper is devoted to the fact that a quasi-uniformity is defined to any topological space, thus to any topological semigroup.

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F -distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.

In this paper, the concept of strongly continuous posets (SC-posets, for short) is introduced. A new intrinsic topology—the local Scott topology is defined and used to characterize SC-posets and weak monotone convergence spaces. Four notions of continuity on posets are compared in detail and some subtle counterexamples are constructed. Main results are: (1) A poset is an SC-poset iff its local Scott topology is equal to its Scott topology and is completely distributive iff it is a continuous precup; (2) For precups, PI-continuity, LC-continuity, SC-continuity and the usual continuity are equal, whereas they are mutually different for general posets; (3) A T 0 -space is an SC-poset equipped with the Scott topology iff the space is a weak monotone convergence space with a completely distributive topology contained in the local Scott topology of the specialization order.

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact , and a topological dual, T k , to obtain the following asymmetric Ellis theorem which applies to the example above: Whenever ( X , ⋅ , T ) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both ( X , ⋅ , T ) and ( X , ⋅ , T k ) , and inversion is a homeomorphism between ( X , T ) and ( X , T k ) . This generalizes the classical Ellis theorem, because T = T k when ( X , T ) is locally compact Hausdorff.

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are: (1) A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism; (2) A poset is continuous iff its Scott topology is completely distributive; (3) A topological T 0 space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology; (4) A topological T 1 space is a discrete space iff its topology is completely distributive. These results generalize the relevant results obtained by J.D. Lawson for dcpos.

A quasi-metric space ( X , d ) is called sup-separable if ( X , d s ) is a separable metric space, where d s ( x , y )=max{ d ( x , y ), d ( y , x )} for all x , y ∈ X . We characterize those preferences, defined on a sup-separable quasi-metric space, for which there is a semi-Lipschitz utility function. We deduce from our results that several interesting examples of quasi-metric spaces which appear in different fields of theoretical computer science admit semi-Lipschitz utility functions. We also apply our methods to the study of certain kinds of dynamical systems defined on quasi-metric spaces.

We give several characterizations of maximal point spaces of bounded continuous dcpos. Among them: they are regular second countable spaces which arise from a quasimetric whose dual gives rise to a compact space. In [2], we use one of these characterizations to show that the Polish spaces are the maximal point spaces of bounded continuous dcpos.

The quasiorders on a setXare equivalent to the topologies onXwhich are closed under arbitrary intersections. We consider the quaisorders onXto be partial orders on the blocks of a partition ofXand use this approach to survey some fundamental results on the lattice of quasiorders onX.

The purpose of this paper is to study which quasi-metrizable spaces are monotonically normal. In particular, we provide a sufficient condition for a quasi-metrizable space to be monotonically normal. This enables us to prove the monotone normality of a certain amount of interesting examples of quasi-metric spaces; for instance, we show that the continuous poset of formal balls of a metric space, endowed with the Scott topology, is a monotonically normal quasi-metrizable space. © 2011 Elsevier B.V.

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.In [KKM] we introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends our previous work by considering some questions associated with image processing:How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? Our construction gives a unified answer to these questions.The crucial step in making our approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits us to define path, arc, and curve as certain continuous functions on such a parameter interval.

We show that the operator which associates the upper topologyto a pre-order relation can be extended to the morphisms of a subcategory of Top, in such way that it results a functor and it is right adjoint of the respective extension of the operator which associates to each topology its specialization pre-order.

Se introduce la categoría de los espacios topológicos consistentes y se muestra que el funtor que, a cada espacio topológico consistente, le asocia el conjunto ordenado por su orden de especialización, tiene como adjunto a izquierda al funtor que, a cada conjunto ordenado, le asocia el conjunto dotado con su topología de Scott, y como adjunto a derecha el funtor que, a cada conjunto ordenado, le asocia el conjunto dotado con su topología débil. We introduce the category of consistent topological spaces and we show that the [unctor which associates lo a consistent topological space the set ordered by its specialization order has as lejt adjoint the functor which associates to a partial ordered set the set endowed with its Scott topology, and as right adjoint the functor which associates to a partial ordered set the set endowed with its weak topology.

Dato il reticolo di tutte le topologie definibili per un insieme infinito X, quelle che sono minime rispetto alla proprietà $T_{FA}$ possono essere identificate. L 'argomento qui presentato offre un approccio teorico ricorrendo ad una relazione di pre-ordine indotta su X dalla topologia data. Conseguentemente viene illustrata la tecnica per stabilire il minimo in questione fornendo una descrizione altemativa della struttura topologica minima. Più precisamente, la struttura minima può essere descritta in rapporto al comportamento della relazione binaria venutasi a creare ed alla topologia intrinseca indotta su di un insieme parzialmente ordinato. Given the lattice of all topologies definable for an infinite set X, those which are minimal with respect to the property $T_{FA}$ are identified. The argument presented offers an approach which may readily be interpreted order-theoretically, by invoking the specialization pre-order induced on X by the given topology. Accordingly, the potential of the technique developed to establish minimality is illustrated in providing an alternative description of the topologically established minimal structure. Specifically, the minimal structure may be described in terms of the behaviour of the naturally occunring specialization order and the intrinsic topology on the resulting partially ordered set.

We consider how properties of the bonding maps of the inverse spectrum determine properties of the inverse limit. Specifically, we study the limits of inverse spectra of finite T0-spaces with bonding maps which are either chaining or normalizing. We will show that if the bonding maps are normalizing, then the inverse limit is a normal T0-space, and therefore, its Hausdorff reflection is its subset of specialization minimal elements. If the maps are chaining, then the inverse limit is a completely normal spectral space; such spaces have been studied since they include the real spectra of commutative rings [C.N. Delzell, J.J. Madden, J. Algebra 169 (1994) 71], and the prime spectrum of a ring of functions, Spec(C(X)). The existence and importance of this class of non-Hausdorff, normal topological spaces was extremely surprising to us. Further, each of these results is reversible; if the inverse limit is normal, then each space in the spectrum is preceded by one whose bonding map to it is normalizing. By way of contrast, the inverse limit of finite T0-spaces with separating bonding maps need not be a normal topological space (Example 3.8(a)) and furthermore, if the spaces of the inverse spectrum are normal, then the Hausdorff reflection of the limit must be zero-dimensional (Theorem 3.15).