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In [R. Heckmann, K. Keimel, Quasicontinuous Domains and the Smyth Powerdomain, Electronic Notes in Theoretical Computer Science 298 (2013), 215–232], Heckmann and Keimel proved that a dcpo P is quasicontinuous iff the poset Fin P of nonempty finitely generated upper sets ordered by reverse inclusion is continuous. We generalize this result to general topological spaces in this paper. More precisely, for any T 0 space ( X , τ ) and U ∈ τ , we construct a topology τ F generated by the basic open subsets U F = { ↑ F ∈ Fin X : F ⊆ U } . It is shown that a T 0 space ( X , τ ) is a hypercontinuous lattice iff τ F is a completely distributive lattice. In particular, we prove that if a poset P satisfies property DINTop, then P is quasi-hypercontinuous iff Fin P is hypercontinuous.

In this paper, the concept of s 2 -quasialgebraic posets is introduced. The main results are: (1) A poset is an s 2 -quasialgebraic iff the σ 2 -topology is a hypercontinuous and algebraic lattice; (2) A poset is s 2 -algebraic iff it is meet s 2 -continuous and s 2 -quasialgebraic.

In this note, we introduce the concepts of weak finitely regular relations and weak hypercontinuous posets. It is proved that when a binary relation $$\rho :X/\rightarrow Y$$ satisfies a certain condition, $$\rho $$ is weak finitely regular if and only if $$(\varphi _{\rho }(X,Y),\subseteq )$$ is a weak hypercontinuous poset. For a poset P, we get that the relation $$\not \le $$ on P is weak finitely regular if and only if P is weak hypercontinuous.

Given any subset selection $\mathcal {Z}$ for posets, we study two weakenings of the known concept of $\mathcal {Z}$-predistributivity, namely, $\mathcal {Z}$-quasidistributivity and $\mathcal {Z}$-meet-distributivity. The former generalizes quasicontinuity, and the latter meet-continuity of complete lattices. We show for global completions $\mathcal {Z}$ that the $\mathcal {Z}$-quasidistributive and $\mathcal {Z}$-meet-distributive posets are the $\mathcal {Z}$-predistributive ones. For the $\mathcal {Z}$-Δ-ideal completion $\mathcal {Z}^{\Delta } P = \{ Y\subseteq P: {\Delta }^{\mathcal {Z}}Y = Y\}$, $\mathcal {P}$-quasidistributivity is $\mathcal {Z}$-quasidistributivity plus $\mathcal {Z}^{\Delta }$-quasidistributivity, provided ${\Delta }^{\mathcal {Z}}$ is idempotent. For $\mathcal {Z}$-continuous normal completions e : P → N, we show that $\mathcal {Z}$-quasidistributivity of P implies that of N, and the converse holds as well if e is $\mathcal {Z}$-initial. This supplements the corresponding results, due to Erne, on the completion-invariance of $\mathcal {Z}$-predistributivity and $\mathcal {Z}$-meet-distributivity. If $\mathcal {Z}$ is a subset system and the $\mathcal {Z}$-below relation on the subsets of a poset P has the interpolation property then P is $\mathcal {Z}$-quasidistributive and may be embedded in a cube by a map that is $\mathcal {Z}^{\Delta }$-continuous and continuous for the lower topologies.

As a common generalization of s 2 -quasicontinuous posets and quasi Z-continuous domains, the concept of s Z -quasicontinuous posets is introduced and some of their basic properties are investigated. It is proved that if a subset system Z satisfies certain conditions, and P is an s Z -quasicontinuous poset, then the Z-way below relation ≪ Z on P has the interpolation property, the space ( P , σ Z ( P ) ) is locally compact and the space ( P , λ Z ( P ) ) is a pospace. It is also proved that under some conditions, a poset is s Z -continuous if and only if it is meet s Z -continuous and s Z -quasicontinuous.

The notion of generalized completely distributivity is modified in order to obtain a tractable and consistent property of ordered sets.

In this paper, we investigate some basic properties, especially categorical properties, of monotone determined spaces. For a topology τ , we construct a monotone determined topology m d ( τ ) . The main results are: (1) for a space ( X , τ ) , then m d ( τ ) is the weakest monotone determined topology on X containing τ ; (2) the category Top md of monotone determined spaces with continuous maps is fully co-reflexive in the category Top of all topology spaces with continuous maps; (3) the category Top md is cartesian closed.

In this paper, the concepts of C-precontinuous posets, quasi C-precontinuous posets and meet Cprecontinuous posets are introduced. The main results are: (1) A complete semilattice P is C-precontinuous (resp., quasi C-precontinuous) if and only if its normal completion is a C-continuous lattice (resp., quasi C-continuous lattice); (2) A poset is both quasi C-precontinuous and Frink quasicontinuous if and only if it is generalized completely continuous; (3) A complete semilattice is meet C-precontinuous if and only if its normal completion is meet C-continuous; (4) A poset is both quasi C-precontinuous and meet C-precontinuous if and only if it is C-precontinuous.

We first introduce and study two new classes of subsets in $T_0$ spaces - Rudin sets and $\wdd$ sets lying between the class of all closures of directed subsets and that of irreducible closed subsets. Using such subsets, we define three new types of topological spaces - $\mathsf{DC}$ spaces, Rudin spaces and $\wdd$ spaces. The class of Rudin spaces lie between the class of $\wdd$ spaces and that of $\dc$ spaces, while the class of $\dc$ spaces lies between the class of Rudin spaces and that of sober spaces. Using Rudin sets and $\wdd$ sets, we formulate and prove a number of new characterizations of well-filtered spaces and sober spaces. For a $T_0$ space $X$, it is proved that $X$ is sober if{}f $X$ is a well-filtered Rudin space if{}f $X$ is a well-filtered $\mathsf{WD}$ space. We also prove that every locally compact $T_0$ space is a Rudin space, and every core compact $T_0$ space is a $\wdd$ space. One immediate corollary is that every core compact well-filtered space is sober, giving a positive answer to Jia-Jung problem. Using $\wdd$ sets, we present a more directed construction of the well-filtered reflections of $T_0$ spaces, and prove that the products of any collection of well-filtered spaces are well-filtered. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and structures.

We prove that (1) for any complete lattice $L$, the set $\mathcal{D}(L)$ of all nonempty saturated compact subsets of the Scott space of $L$ is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice $L$ is non-sober, then the Scott space of $\mathcal{D}(L)$ is non-sober. Using these results and the Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus give a positive answer to a problem posed by Jung. We will also prove that a $T_0$ space is well-filtered iff its upper space (the set $\mathcal{D}(X)$ of all nonempty saturated compact subsets of $X$ equipped with the upper Vietoris topology) is well-filtered, which answers another open problem., Comment: 9 pages

Let Top d , Top w and Sob be the category of all d-spaces, that of all well-filtered spaces and that of all sober spaces respectively. For a full subcategory K of Top d containing Sob, it is proved that the product of an arbitrary family of K-determined sets is a K-determined set and if K is adequate, then the K-reflection preserves arbitrary products of T 0 spaces. In particular, the Keimel-Lawson reflection, well-filtered reflection and d-reflection all preserve arbitrary products of T 0 spaces, and DCPO-completion preserves the product of a family { P i : i ∈ I } of posets if the Scott topology of the product ∏ i ∈ I P i is the product of the Scott topologies of the factors.

In this paper, we consider a common generalization of both s 1 -continuous posets and quasicontinuous domains, and we introduce new concepts of way below relations and s 1 -quasicontinuous posets. The main results are: (1) A poset is an s 1 -quasicontinuous poset iff the σ 1 -topology is a hypercontinuous lattice iff the S ⁎ -convergence is topological with respect to the σ 1 -topology; (2) A poset is s 1 -continuous iff it is meet s 1 -continuous and s 1 -quasicontinuous; (3) The λ 1 -topology on an s 1 -quasicontinuous poset is Tychonoff; (4) A poset P is s 1 -quasicontinuous and the σ 1 -topology is sober iff P is a quasicontinuous domain and the σ 1 -topology coincides with the Scott topology.

In this note it is proved that for a quasicontinuous lattice L, the lower topology ω ( L ) and the Scott topology σ ( L ) are duals for each other; and if L is a complete lattice such that σ ( L ) is continuous but not hypercontinuous (equivalently, L is not quasicontinuous), then ω ( L ) is not the dual of σ ( L ) and hence they are not duals for each other.

We first introduce and study two new classes of subsets in T 0 spaces — ω-Rudin sets and ω-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces — ω-d-spaces and ω-well-filtered spaces. We prove that an ω-well-filtered T 0 space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable ω-well-filtered T 0 space are directed. Therefore, a first countable T 0 space X is sober iff X is well-filtered iff X is an ω-well-filtered d-space. Using ω-well-filtered determined sets, we present a direct construction of the ω-well-filtered reflections of T 0 spaces, and show that products of ω-well-filtered spaces are ω-well-filtered.

In this paper, we provide a direct approach to $\mathbf{K}$-reflections of $T_0$ spaces. For a full subcategory $\mathbf{K}$ of the category of all $T_0$ spaces and a $T_0$ space $X$, let $\mathbf{K}(X)=\{A\subseteq X : A$ is closed and for any continuous mapping $f : X\longrightarrow Y$ to a $\mathbf{K}$-space $Y$, there exists a unique $y_A\in Y$ such that $\overline{f(A)}=\overline{\{y_A\}}\}$ and $P_H(\mathbf{K}(X))$ the space of $\mathbf{K}(X)$ endowed with the lower Vietoris topology. It is proved that if $P_H(\mathbf{K}(X))$ is a $\mathbf{K}$-space, then the pair $\langle X^k=P_H(\mathbf{K}(X)), \eta_X\rangle$, where $\eta_X :X\longrightarrow X^k$, $x\mapsto\overline{\{x\}}$, is the $\mathbf{K}$-reflection of $X$. We call $\mathbf{K}$ an adequate category if for any $T_0$ space $X$, $P_H(\mathbf{K}(X))$ is a $\mathbf{K}$-space. Therefore, if $\mathbf{K}$ is adequate, then $\mathbf{K}$ is reflective in $\mathbf{Top}_0$. It is shown that the category of all sober spaces, that of all $d$-spaces, that of all well-filtered spaces and the Keimel and Lawson's category are all adequate, and hence are all reflective in $\mathbf{Top}_0$. Some major properties of $\mathbf{K}$-spaces and $\mathbf{K}$-reflections of $T_0$ spaces are investigated. In particular, it is proved that if $\mathbf{K}$ is adequate, then the $\mathbf{K}$-reflection preserves finite products of $T_0$ spaces. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and their categorical structures., Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1909.09303

In this paper, the concept of meet F-continuous posets is introduced. The main results are: (1) A poset P is meet F-continuous iff its normal completion is a meet continuous lattice iff a certain system γ(P) which is, in the case of complete lattices, the lattice of all Scott closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff P is meet F-continuous and quasiprecontinuous; (3) The category of meet continuous lattices with complete homomorphisms is a full reflective subcategory of the category of meet F-continuous posets with cut-stable maps.

The concepts of hypercontinuous posets and generalized completely continuous posets are introduced. It is proved that for a poset P the following three conditions are equivalent: (1) P is hypercontinuous; (2) the dual of P is generalized completely continuous; (3) the normal completion of P is a hypercontinuous lattice. In addition, the relational representation and the intrinsic characterization of hypercontinuous posets are obtained.

In this short note, we prove that a complete lattice is a generalized completely distributive lattice if and only if its order dual is a hypercontinuous lattice.

In this paper the classical theorem of Zareckiĭ about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckiĭ theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the Urysohn-Nachbin lemma is presented which is quite different from the classical one.

In this paper, we introduce the concept of meet precontinuous posets, a generalization of meet continuous lattices to posets. The main results are: (1) A poset P is meet precontinuous iff its normal completion is a meet continuous lattice iff a certain system γ(P) which is, in the case of complete lattices, the lattice of all Scott-closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff the way below relation is the smallest approximating auxiliary relation iff P is meet precontinuous and there is a smallest approximating auxiliary relation on P. Finally, given a poset P and an auxiliary relation on P, we characterize those join-dense subsets of P whose way-below relation agrees with the given auxiliary relation.

In this paper, concepts of strongly way below relations, completely precontinuous posets, coprimes and Heyting posets are introduced. The main results are: (1) The strongly way below relations of completely precontinuous posets have the interpolation property; (2) A poset P is a completely precontinuous poset iff its normal completion is a completely distributive lattice; (3) An ω-chain complete P is completely precontinuous iff P and Pop are precontinuous and its normal completion is distributive iff P is precontinuous and has enough coprimes; (4) A poset P is completely precontinuous iff the strongly way below relation is the smallest approximating auxiliary relation on P iff P is a Heyting poset and there is a smallest approximating auxiliary relation on P. Finally, given a poset P and an auxiliary relation on P, we characterize those join-dense subsets of P whose strongly way-below relation agrees with the given auxiliary relation.