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𝒵 -distributive function lattices

Marcel Erné (2013)

Mathematica Bohemica

It is known that for a nonempty topological space X and a nonsingleton complete lattice Y endowed with the Scott topology, the partially ordered set [ X , Y ] of all continuous functions from X into Y is a continuous lattice if and only if both Y and the open set lattice 𝒪 X are continuous lattices. This result extends to certain classes of 𝒵 -distributive lattices, where 𝒵 is a subset system replacing the system 𝒟 of all directed subsets (for which the 𝒟 -distributive complete lattices are just the continuous...

A metrizable completely regular ordered space

Hans-Peter A. Künzi, Stephen W. Watson (1994)

Commentationes Mathematicae Universitatis Carolinae

We construct a completely regular ordered space ( X , 𝒯 , ≤ ) such that X is an I -space, the topology 𝒯 of X is metrizable and the bitopological space ( X , 𝒯 ♯ , 𝒯 ♭ ) is pairwise regular, but not pairwise completely regular. (Here 𝒯 ♯ denotes the upper topology and 𝒯 ♭ the lower topology of X .)

A principal topology obtained from uninorms

Funda Karaçal, Tuncay Köroğlu (2022)

Kybernetika

We obtain a principal topology and some related results. We also give some hints of possible applications. Some mathematical systems are both lattice and topological space. We show that a topology defined on the any bounded lattice is definable in terms of uninorms. Also, we see that these topologies satisfy the condition of the principal topology. These topologies can not be metrizable except for the discrete metric case. We show an equivalence relation on the class of uninorms on a bounded lattice...

Domain representability of C p ( X )

Harold Bennett, David Lutzer (2008)

Fundamenta Mathematicae

Let C p ( X ) be the space of continuous real-valued functions on X, with the topology of pointwise convergence. We consider the following three properties of a space X: (a) C p ( X ) is Scott-domain representable; (b) C p ( X ) is domain representable; (c) X is discrete. We show that those three properties are mutually equivalent in any normal T₁-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk that C p ( X ) is...

Domain-representable spaces

Harold Bennett, David Lutzer (2006)

Fundamenta Mathematicae

We study domain-representable spaces, i.e., spaces that can be represented as the space of maximal elements of some continuous directed-complete partial order (= domain) with the Scott topology. We show that the Michael and Sorgenfrey lines are of this type, as is any subspace of any space of ordinals. We show that any completely regular space is a closed subset of some domain-representable space, and that if X is domain-representable, then so is any G δ -subspace of X. It follows that any Čech-complete...

Embeddings of chains into chains

Vítězslav Novák, Tomáš Novotný (2005)

Discussiones Mathematicae - General Algebra and Applications

Continuity of isotone mappings and embeddings of a chain G into another chain are studied. Especially, conditions are found under which the set of points of discontinuity of such a mapping is dense in G.

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