General Topology Papers of the Week

To this end, we prove that, for every topological MV-algebra, the associated Mundici group can be endowed with a topology that extends the topology of the MV-algebra and makes it into an abelian topological $\ell$-group.

We give two natural Hofmann--Mislove representations in the general $H$-sobriety framework: filters induced by $H$-sets of $X$ characterize $H$-sobriety, whereas filters induced by $H$-sets of the Smyth power space characterize super $H$-sobriety.

Our main result shows that, under the assumption of tautness, the same phenomenon holds at the categorified level, thereby offering a non-ultra-biased perspective on virtual ultracategories.

Given a subset system $H$ on $T_0$-spaces, we associate with each topology a closure-testing derived topology and study the fixed points of this operator.

We show that the Scott power construction does not preserve co-sobriety.

We investigate how order structure controls sequential convergence in the lower topology of a poset and how this, in turn, governs topological duality and co-sobriety of Scott spaces.

We classify the $k$-irreducible compact saturated subsets of the extension and show that each is a point saturation.

We give a unified treatment of the four properties $\W$, $\Ww$, $\Wi$ and $\Win$.

For an arbitrary poset $P$, we describe the weakest order-compatible irreducible determined topology $\rho(P)$ using Dedekind--MacNeille cuts and a finite-cover condition involving principal lower sets.

The Wilker property requires that a compact set covered by two open sets admit compact covering pieces inside those open sets. We study this property through products, compact-open function spaces and hyperspaces.

We introduce the notion of Boomerang subalgebras of group von Neumann algebras and study their general properties.

In this paper, we prove that for every topological MV-algebra the topology is induced by a uniformity.

We prove that the indicator functions form a uniformly continuous retract of the set of $[0,1]$-valued functions in $L^2([0,1])$.

We construct topological partial actions on products and disjoint unions of topological spaces, and G-pinvariant subspaces.

In this paper, we focus on well-filtered spaces.

We show that upper semi-continuity arises naturally in a voting model, in which a panel of experts divides a metric universe into closed regions of acceptance and rejection, and for this model we construct an explicit continuous selection, the consensus selection, illustrated by a non-compact example in the plane.

Extending this perspective, we explore how walks in the graph, viewed as sequences of subsets of vertices $V$, fit into a more general framework where the path distance emerges as a special case of broader metrics that define hypertopologies on the space of subsets of $V$. In this paper, we propose a general framework to address this issue, motivated by the extension of recent reinforcement learning algorithms on graphs through Lipschitz function extensions.

We study the relationship between nonpathological lower semicontinuous submeasures, $F_\sigma$ ideals on $\omega$, and matrix summability ideals.

M. Tkachenko asked whether a regular feebly compact semitopological group is a topological group provided that it is separable, has countable tightness, or is a $k$-space; for Tychonoff groups the answer is known to be positive. We answer this question in the separable case: every regular separable feebly compact semitopological group is a topological group.

We prove that under $\mathrm{MA}(\omega_1)$, if all finite powers of $X$ belong to NFS, then $C_p(X)^\omega$ is ESLC; thus, for subspaces of $C_p(X)$, hereditary Lindelöfness, hereditary separability and countable spread coincide.