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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 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.