General Topology

We revisit Talagrand's CH compactum as a test object for the two-disjoint-copies property and for pointwise quotient questions. The two-disjoint-copies property, or 2DCP, is a topological sufficient condition for the existence of…

In this paper, we show that there is a countable Noetherian complete lattice $L$ and an order-compatible $d$-topology $\tau$ on $L$ such that $(L, \tau)$ is not well-filtered, and there exist a dcpo $P$ and an order-compatible well-filtered…

We show that every Lindel\"of scattered $W$-space is $\sigma$-compact. This result generalizes a theorem proved recently by Avil\'es and the author in [Topology Appl. 363 (2025), Paper No. 109234] and answers a question posed by Tkachuk.

A coherent domain, in the sense of Dana Scott's domain theory, is a domain in which the intersection of every two compact saturated subsets is again compact, when the domain is equipped with the Scott topology. Coherence plays key roles in…

In this paper, we study when the Smyth powerspace $\mathcal{Q}^*_v(X)$ of a topological space $X$ is coherent, and prove that $X$ is coherent and weakly Hausdorff if and only if $\mathcal{Q}^*_v(X)$ is coherent and weakly Hausdorff. We give…

We provide details for a Theorem which is only available on a unfinished preliminary draft of P. Nyikos: the existence under $\clubsuit_C$ of a hereditarily collectionwise normal countably compact non-compact manifold which does not contain…

For a finite nonempty poset \(F\), the normalized probabilistic powerdomain \(\Vone(F)\) is an RB-domain exactly when \(F\) is a finite rooted tree. We extend this classification to arbitrary nonempty dcpos from the viewpoint of forbidden…

In this paper, we provide a complete isomorphism classification of the spaces $C_p(K)$ of real-valued continuous functions endowed with the topology of pointwise convergence for separable compact lines $K$ of weight $\omega_1$, under the…

Let $C$ be a closed, convex, pointed and generating cone in a finite-dimensional real vector space $V$, and let \( D_C=(-C)\cup\{\bot\}\) be the negative cone with a new least element, ordered by the cone order. Keimel proved that these…

We study linear topologies on residuated lattices generated by systems of filters, with emphasis on the uniform structures and separation properties that they determine. A down-directed family of filters gives a natural compatible…

We present a solution to some problems posed by the author and Kalenda. We show that the closed ball of nonseparable Hilbert space in its weak topology is homeomorphic to its positive part, as well as to its product with the Hilbert cube.…

A topological space \(X\) is called a \(Q\)-space if every subset of \(X\) is a \(G_\delta\)-set, and \(X\) is a \(\Delta\)-space if for any decreasing sequence \(\{D_n : n \in\omega\}\) of subsets of \(X\) with empty intersection there is…

We prove that Lawson's planar closed-disk domain is not an RB-domain. This domain is the dcpo of all closed disks in the Euclidean plane, together with the whole plane as bottom, ordered by reverse inclusion. Since this domain is an…

Recent advances in biomedicine generate high-dimensional single-cell data that describe cellular heterogeneity with unprecedented detail, but their geometric complexity and non-linear structure often limit the effectiveness of conventional…

For a given Tychonoff space $X$, a point $p\in \beta(X)\setminus X$ is called {\em remote} if $p$ is not in the closure of any nowhere dense subset of $X$. In this paper, we characterize spaces with remote points in terms of certain…

A Tychonoff space $X$ has the two-disjoint-copies property (2DCP) if there exists a sequence $(K_n)_{n\in\omega}$ of non-empty compact subsets of $X$ such that each $K_n$ contains two disjoint subsets homeomorphic to $K_{n+1}$. Banakh,…

In this short note we give a negative answer to the following open question: \emph{Let $X$ be a $\sigma$-compact paratopological group. Does there exist a continuous isomorphism of $X$ onto a topological group $G$?} Specifically, we…

We investigate the local topological structure of non-metrizable topological groups through the lens of Tukey order and cofinal types. Motivated by recent advances in topological groups admitting an $\omega^\omega$-base, we introduce the…

Let $G$ be a compact group. The existence of certain $G$-homotopy dense subsets in a metrizable $G$-space $X$ plays a fundamental role, as it is equivalent to $X$ being a $G$-ANR. From this perspective, the present paper develops several…

K\"unzi and Yildiz introduced convexity structures in the sense of Takahashi for $T_{0}$-quasi-metric spaces. In this article, we continue this line of study on the Isbell-convex hull of an asymmetrically normed real vector space. Using the…