Abstract We first introduce and investigate a new class of $T_0$ -spaces – strong $R$ -spaces, which are stronger than both $R$ -spaces and strongly well-filtered spaces. It is proved that any sup-complete poset equipped with the upper topology is a strong $R$ -space, and the Hoare power space of a $T_0$ -space is a strong $R$ -space. Hence, the upper topology on a sup-complete poset is strongly well-filtered, and the Hoare power space of a $T_0$ -space is strongly well-filtered, which answers two problems recently posed by Xu.

Abstract The author’s primary goal in this paper is to characterize $\omega$ -Rudin sets and $\omega$ -Rudin spaces via sequence convergence and give some important applications of such characterizations. For an irreducible closed set $A$ of a $T_0$ -space $X$ , we prove that the following four conditions are equivalent: (1) $A$ is an $\omega$ -Rudin set; (2) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(a_n)_{n\in \mathbb{N}}$ simultaneously converges to all points of $A$ ; (3) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the Hoare power space of $X$ ; (4) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the sobrification of $X$ . Based on these characterizations, we obtain some characterizations of $\omega$ -Rudin spaces and sober spaces. In particular, we show that for a complete lattice $L$ , its Scott space $\Sigma L$ is sober iff for any nonempty Scott irreducible closed set $A$ of

Abstract The authors’ primary goal in this paper is to enhance the study of $T_0$ topological spaces by using the order of specialization of a $T_0$ -space to introduce the lower topology (with a subbasis of closed sets $\mathord{\uparrow } x$ ) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets $\mathord{\uparrow } F$ , $F$ finite, is contained in an open set $U$ , then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong $d$ -spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo $P$ with the Scott topology is a strong $d$ -space, then it is well-filtered, and if additionally the Scott topology of the product $P\times P$ is the product of the Scott topologies of the factors, then the Scott space of $P$ is sober. We also exhibit connections of this work with de Groot duality.

Transcutaneous auricular vagus nerve stimulation (taVNS) is an emerging neuromodulation technology that has been reported to be beneficial in the treatment of diseases by several studies, but its exact mechanism of action is still unclear. It has been demonstrated that taVNS can influence interoceptive signals. Notably, the processing of interoceptive signals is directly related to many diseases, such as depression, anxiety, and insomnia. The insula and the medial prefrontal cortex (MPFC) communicate during the bottom-up transmission of taVNS-induced signals, and both play a role in interoceptive signal processing. By focusing on the insula and MPFC, our research pioneers detail the potential interactions between interoceptive signal processing and the neuromodulation effects of taVNS, providing novel insights into the neurobiological mechanisms of taVNS. Two functional connectivity (FC) analyses (region of interest-based and seed-based) were used in this study. We observed that negative connectivity between the insula and the MPFC was significantly weakened following taVNS, while there were no statistical changes in the sham group. Our findings elucidate potential mechanisms linking vagal activity with intrinsic FC among specific brain regions and networks. Specifically, our results indicate that taVNS may enhance the ability to flexibly balance interoceptive awareness and cognitive experiences by modulating the FC between the insula and MPFC. The modulation effects may impact body-brain interactions, suggesting the mechanism of taVNS in therapeutic applications.

Abstract For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$ . $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$ ) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X, its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$ . A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces, and the Scott topology on Smyth power posets, are also investigated.

Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $\Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for $\Psi$ -well-filtered spaces is provided via $\Psi$ -sets. We also discuss the relationship between $\Psi$ -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete $\Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $\Psi$ that has a designated property, we show that a $T_0$ space X is $\Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $\Psi$ -well-filtered.

Abstract In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on $\mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X, Y is a d-space (resp., well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $\Sigma \mathcal O(X)$ is non-sober, the function space $\mathbb{C}(X, \Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $\mathbb{C}(X, Y)$ equipped with the Scott topology, the compact-open topology and the pointwise convergence topology are also discussed. Our study also leads to a number of questions, whose answers will deepen our understanding of the function spaces related to H-sober spaces.

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.