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Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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 -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. [Copyright &y& Elsevier]

Abstract: In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology 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, , to obtain the following asymmetric Ellis theorem which applies to the example above: Whenever is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both and , and inversion is a homeomorphism between and . This generalizes the classical Ellis theorem, because when is locally compact Hausdorff. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to investigate the relationships between certain generalized topological structures and fuzzy order structures on a universe set X. In details, a fuzzy preordered relation induces an I-fuzzy Alexandrov topology; and an I-fuzzy topology induces a fuzzy preordered relation, which turns out to be a specialization order in I-fuzzy setting. In addition, the Representation Theorem of fuzzy preorders by I-fuzzy topologies is obtained. These results coincide with the conclusions obtained in literature in the version of two-valued logic. [Copyright &y& Elsevier]

Abstract: This paper presents, by means of Alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces, a systematic investigation of the interrelationship between fuzzy preordered sets, topological spaces, and fuzzy topological spaces. [Copyright &y& Elsevier]

Abstract: 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 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 space is a discrete space iff its topology is completely distributive. These results generalize the relevant results obtained by J.D. Lawson for dcpos. [Copyright &y& Elsevier]

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). [Copyright &y& Elsevier]

In this paper, we continue the study of the cancellation problem for the cartesian product of partially ordered sets (posets for short) and of T 0 -spaces. We first give a class of partially ordered sets satisfying the cancellation law which is different from that investigated by Banaschewski and Lowen, and then give its corresponding topological version. [ABSTRACT FROM AUTHOR]