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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, we construct a new way below relation from any given T 0 space by making use of the cut operator, and introduce the concepts of SI2-continuous spaces and SI2-quasicontinuous spaces. The main results are: (1) a space ( X , τ ) is SI2-continuous iff the set X equipped with the SI2-topology τ S I 2 is a C-space; (2) a space ( X , τ ) is SI2-quasicontinuous iff ( X , τ S I 2 ) is a locally hypercompact space; (3) a space is SI2-continuous iff it is a meet SI2-continuous and SI2-quasicontinuous space.

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 paper, we mainly investigate some basic properties of almost sober spaces. For the category ASob of all almost sober spaces with continuous mappings, it is proved that the ASob-reflection of Johnstone space Σ J does not exist and hence ASob is not reflective in the category Top 0 of all T 0 spaces with continuous mappings. It is shown that some properties which are similar to that of sober spaces hold and others do not hold.

In [1] , Escardo, Lawson and Simpson introduced the concept of co-sober spaces and asked whether a sober space is co-sober. In this paper, we answer this question in the negative by a counterexample. The other main results are: (1) A closed subspace of a co-sober space is co-sober; (2) A saturated subspace of a co-sober space is co-sober; (3) If ( X , τ ) is a co-sober space and a continuous mapping c : ( X , τ ) → ( X , τ ) is a closure operator with respect to the specialization preorder of ( X , τ ) , then the subspace ( c ( X ) , τ | c ( X ) ) is a co-sober space; (4) If the Smyth power space P s ( X ) of a topological space X is co-sober, then X is co-sober.

In this paper, the concepts of g s 2 -convergence and quasi-liminf convergence of filters in posets are introduced. The main results are: (1) For an order consistent topology τ on a poset P, P is τ-quasicontinuous iff the g s 2 -convergence coincides with τ-convergence; (2) For an order consistent topology τ on a poset P, the quasi-liminf convergence coincides with τ ∨ ω ( P ) -convergence if P is τ-quasicontinuous; (3) For an order consistent topology τ on a poset P, P is τ-continuous iff the s 2 -convergence coincides with τ-convergence iff it is meet τ-continuous and the quasi-liminf convergence coincides with τ ∨ ω ( P ) -convergence.

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.

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.

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

Following Ershov's method of constructing the d-completion of T 0 spaces, we give a direct construction of the well-filtered reflection of T 0 spaces. Also, we obtain an elegant characterization of well-filtered spaces using KF-sets. We then show that a product of a family of T 0 spaces is well-filtered iff each factor space is well-filtered. Finally, we obtain that the well-filtered reflection of a product of a finite family of T 0 spaces is the product of the well-filtered reflections of all factor spaces. A common theme is that KF-sets, introduced by the first and the fourth authors, function prominently in all of the above proofs.