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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 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.

The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an {\omega}{\omega}-base and G is Frechet-Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn and having an {\omega}{\omega}-base are all weakly three-space properties with H a closed L-subgyrogroup, Comment: 10 pages

Anaplastic thyroid cancer (ATC) is a highly aggressive and the most lethal type of thyroid cancer. The standard-of-care for unresectable ATC is radiotherapy and chemotherapy, usually based on doxorubicin (Dox). However, most patients develop resistance shortly after treatment. To overcome the drug resistance, we synthesized the mesoporous organosilica nanoparticles (MONPs) loaded with Dox and stabilized the nanocomposites by bovine serum albumin (BSA). The surface area and pore volume of MONPs were 612.653 m2/g and 0.589 cm3/g. The loading capacity of Dox-MONPs reached 47.02%. Compared to Dox-MONPs and free Dox, BSA-Dox-MONPs had more durable tumor-killing power on both drug-sensitive cell line HTh74 and drug-resistant cell line HTh74R. The cellular uptake of BSA-Dox-MONPs was 28.14 and 65.53% higher than that of Dox-MONP in HTh74 and HTh74R. Furthermore, the BSA coating decreased the efflux rate of nanocomposites in HTh74 (from 38.95 to 33.05%) and HTh74R (from 43.03 to 32.07%). In summary, BSA-Dox-MONPs reversed the chemotherapy resistance of ATC cells via increased drug uptake and inhibited drug efflux, offering a promising platform for the treatment of chemo-resistant ATC.

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.

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.

For a $T_0$ space $X$, let $\mk (X)$ be the poset of all compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if for any $K_1, K_2\in \mk (X)$, $K_2\ll K_1$ in $\mk (X)$ if{}f $K_2\subseteq \ii~\!K_1$. In this paper, we give several connections among the well-filteredness of $X$, the sobriety of $X$, the local compactness of $X$, the core compactness of $X$, the property Q of $X$, the coincidence of the upper Vietoris topology and Scott topology on $\mk (X)$, and the continuity of $x\mapsto\ua x : X \longrightarrow \Sigma~\!\! \mk (X)$ (where $\Sigma~\!\! \mk (X)$ is the Scott space of $\mk (X)$). It is shown that for a well-filtered space $X$ for which its Smyth power space $P_S(X)$ is first-countable, the following three properties are equivalent: the local compactness of $X$, the core compactness of $X$ and the continuity of $\mk (X)$. It is also proved that for a first-countable $T_0$ space $X$ in which the set of minimal elements of $K$ is countable for any compact saturated subset $K$ of $X$, the Smyth power space $P_S(X)$ is first-countable. For the Alexandroff double circle $Y$, which is Hausdorff and first-countable, we show that its Smyth power space $P_S(Y)$ is not first-countable., Comment: 12 pages

In the past few years, the research on sober spaces and well-filtered spaces has got some breakthrough progress. In this paper, we shall present a brief summarising survey on some of such development. Furthermore, we shall pose and illustrate some open problems on well-filtered spaces and sober spaces., Comment: 17 pages. arXiv admin note: text overlap with arXiv:2003.07737, arXiv:1909.09303, arXiv:1911.11618

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.

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.

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.

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.

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.

Based on topological Rudin's Lemma, we investigate two new kinds of sets - Rudin sets and well-filtered determined sets in $T_0$ topological spaces. Using such sets, we formulate and prove some new characterizations for well-filtered spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work on well-filtered spaces. Our study also lead to a new class of spaces - strong $d$-spaces and some problems whose solutions will strengthen our understanding of the related structures., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1909.09303

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

We prove that for every T 0 space X, there is a well-filtered space W ( X ) and a continuous mapping η X : X ⟶ W ( X ) , such that for any well-filtered space Y and any continuous mapping f : X ⟶ Y there is a unique continuous mapping f ˆ : W ( X ) ⟶ Y such that f = f ˆ ∘ η X . Such a space W ( X ) will be called the well-filterification of X. This result gives a positive answer to one of the major open problems on well-filtered spaces. As a corollary, we obtain that the product of well-filtered spaces is well-filtered.

In [3] , Erne relaxed the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing, and introduced three kinds of non-sober spaces: cut spaces, weakly sober spaces, and quasisober spaces. In this paper, their basic properties are investigated. It is shown that some properties which are similar to that of sober spaces hold and others do not hold.

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.

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.