Papers

In this paper we prove that $B_1(X)$ has a dense cofinally Polish subspace if and only if $X$ is a $\Delta_1$-space. Also we prove that $B_1(X, [0,1])$ has a dense cofinally Polish subspace if and only if $X$ is a weak $\lambda$-space. We also establish that $C_p(X, [0,1])$ is pseudocompact if and only if $C_p(X,[0,1])$ has a dense cofinally Polish subspace. This result provides answers to four open questions from (Tkachuk in RACSAM 115(68), 2021).

Strong well-filteredness is a refinement of well-filteredness arising naturally in non-Hausdorff topology and domain theory. In this paper, we construct a countable locally compact strong R-space X such that neither its Smyth power space nor its Scott power space is a strong d-space. This example gives negative answers to the questions concerning the preservation of strong well-filteredness and the strong R-space property under power space constructions. Furthermore, for the two-point discrete space \two, the function space (\two,X) is not a strong d-space under any of the pointwise convergence, compact-open, Isbell, core-open or Scott topologies, where the Scott topology is induced by the pointwise order.

We answer in ZFC a question of N. Carlson by constructing a zero-dimensional Tychonoff space $X$, with exactly one non-isolated point, for which every point admits a decreasing local $\pi$-base and \[ \pi\chi(X)=\pi\chi_{\mathrm l}(X)=\omega <\pi\chi_{\mathrm c}(X)=\omega_1 <\pi\chi_{\mathrm d}(X)=\omega_2. \] A preliminary chain-refinement lemma shows that every inclusion-chain of subsets of a set has a reverse-well-ordered coinitial subchain of controlled cardinality. Thus the result is independent of whether a decreasing local $\pi$-base is understood as an inclusion-chain, a strictly decreasing transfinite sequence, or a weakly decreasing transfinite sequence. The construction combines a countable star filter, whose centred local $\pi$-character is at least the pseudointersection number $\mathfrak p$, with a matrix filter that gives the exact values $\omega_1$ and $\omega_2$.

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

Let \(H(A)\) be the Hattori space associated with \(A\subseteq\mathbb R\), and put \(B=\mathbb R\setminus A\). Kulesza announced at the 2018 Joint Mathematics Meetings that Lindelöfness of \(H(A)^{2n}\) forces Lindelöfness of \(H(A)^{2n+1}\) for every positive integer \(n\). The meeting abstract states this implication without a proof. This paper proves the case \(n=1\) in ZFC. The same one-sided geometry gives a pointwise half-space criterion for affine discreteness and, when \(H(A)^2\) is Lindelöf, a four-coordinate obstruction for uncountable closed discrete monotone traces. Known results on mixed Sorgenfrey–Euclidean products lead to hereditary-Lindelöf criteria for finite powers, a countable-power characterization, and a finite-power Lindelöf reduction. A transfinite construction gives \(A\subseteq\mathbb R\) for which \(H(A)^2\) and \(H(A)^3\) are Lindelöf, whereas \(H(A)^4\) contains a closed discrete affine line of cardinality \(\mathfrak c\).

A dcpo \(D\) is called \(C_{\sigma}\)-unique if for every dcpo \(Q\), \( \Gamma(D)\cong \Gamma(Q) \) implies \( D\cong Q, \) where \(\Gamma(D)\) denotes the lattice of Scott-closed subsets of \(D\). We give an intrinsic characterization of \(C_{\sigma}\)-uniqueness in terms of Skula-density and Scott closure, identifying precisely when a proper sub-dcpo can preserve the lattice of Scott-closed sets. Based on this characterization, we answer three open problems: (1) every power $\J^I$ of Johnstone's dcpo is $C_{\sigma}$-unique; (2) $C_{\sigma}$-uniqueness is not preserved by binary products, even when both factors and their product are sober; and (3) a sober countable complete lattice need not be $C_{\sigma}$-unique, even when it is a frame.

A metric space is plastic if every bijective nonexpansive self-map is an isometry. We prove that the hyperspace of nonempty compact subsets of every connected, locally finite, regular graph, equipped with the Hausdorff metric associated with the path metric, is plastic. The proof identifies the sets with smallest unit balls as the nonempty subsets of adjacent-twin classes and shows that every nonexpansive bijection induces an automorphism of the quotient graph preserving the sizes of these classes. Singletons are preserved when there are no adjacent twins, but need not be preserved in general. We also prove that $\mathcal{K}(K\times G)$ is plastic for every compact connected metric space $K$ and every connected, locally finite, regular graph $G$, with the supremum metric on the product, and give a more general criterion for such products. Further results include a plastic metric space whose hyperspace is not plastic, plasticity of every tree, and plasticity of the hyperspace of a connected, locally finite graph with only finitely many vertices of minimum degree.

In this paper, we obtain two complete lattices that are sober with respect to the Scott topology, whereas the Scott space of their product is not a sober space. Thus, we answer a question posed by Miao, Xi, Jia, Li and Zhao in [8].

Let $\Sigma_{1} = \mathbb{N}^*$ and $\Sigma_{k+1} = \overline{\mathbb{N}^* + \Sigma_{k}}$ be the cumulative depth filtration of $\beta\mathbb{N}$, the analogue for $(\mathbb{N},+)$ of a chain of closed ideals that Protasov and Protasova studied for discrete groups, where strict descent follows from a theorem of Lutsenko and Protasov. For every $k$ we give an explicit set whose closure meets $\Sigma_{k}$ but not $\Sigma_{k+1}$. Fix the doubly exponential sequence $e_{n} = 2^{2^{n}}$, partition it into $k$ subsequences $E_{0}, \dots, E_{k-1}$ by the residue of the index modulo $k$, and set $A_{k} = E_{0} + \cdots + E_{k-1}$. We prove that any sum $q_{0} + \cdots + q_{k-1}$ of free ultrafilters with $E_{t} \in q_{t}$ lies in $\Sigma_{k} \setminus \Sigma_{k+1}$. The engine is a master lemma, proved by induction on $j$: if a sum $F_{1} + \cdots + F_{j}$ of subsequences of $\{e_{n}\}$ with pairwise disjoint index sets belongs to a free ultrafilter $s$, then $s \notin \Sigma_{j+1}$. The proof rests on a single rigidity of the doubly exponential sequence: a fixed difference forces the largest index in any shift-intersection, once it is large, to cancel within its own subsequence, which makes every shift-intersection descend by at least one level. The same witnesses lie in the gaps of the pure filtration.

For any Tychonoff space $X$, let $D(X)$ denote either the space $C(X)$ of all continuous real-valued functions on $X$ or the space $C^*(X)$ of all bounded continuous real-valued functions on $X$. We write $D_p(X)$ when $D(X)$ is endowed with the topology of pointwise convergence. In our recently published paper [8, Theorem 1.6], we obtained the following result: if $T: D_{p}(X) \to D_{p}(Y)$ is a linear continuous surjection, where $X$ is a metrizable space and $Y$ is a perfectly normal space, then $Y$ inherits a given topological property $\mathcal{P}$ from $X$. A linear continuous surjection $T: E_{p}(X)\to E_{p}(Y)$ is said to be densely defined if $E(X)$ and $E(Y)$ are dense linear subspaces of $D_{p}(X)$ and $D_{p}(Y)$, respectively, and $E(X)$ is correct (see Definition 1.5(a)). In the present paper, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$. In particular, $\mathcal{P}$ can be zero-dimensionality, $\sigma$-compactness or strong countable-dimensionality. Additionally, for arbitrary Tychonoff spaces $X$ and $Y$, assuming that $T: E_p(X)\to E_p(Y)$ is a densely defined linear continuous operator, we show that $X\in\mathcal P$ implies $Y\in\mathcal P$, where $\mathcal P$ is the property $(\kappa)$, the strong $\sigma$-scatteredness, or the property of being a $\Delta_1$-space.

For a general directed space, forgetting the parametrization of paths can change their homotopy type. A clock is a directed space with submetrizable underlying space; a timed space is a saturated directed space equipped with a regular directed map to a clock. Over every fixed clock, timed spaces form a locally presentable category, and their quotients from directed paths with fixed endpoints to traces are trivial Hurewicz fibrations with continuous sections. For Hausdorff saturated directed spaces with metrizable quasicompact subspaces, these quotients are trivial q-fibrations; this includes the second countable case. The directed circle is the cubical clock, but a finite cellular globular example admits no regular map to it. We construct a globular clock from a cellular q-cofibrant replacement of the terminal multipointed $d$-space. It contains the cubical clock by an injective directed map and admits regular maps from all q-cofibrant globular realizations, which are themselves saturated clocks. Thus timed spaces over this clock include both cubical and globular examples. Normalization gives a new proof of the quotient theorem for execution paths of q-cofibrant multipointed $d$-spaces and proves that the quotient by nondecreasing surjections is a trivial Hurewicz fibration with a continuous section.

We employ the set-theoretic methods yielding the consistency of the existence of nontrivial pairwise disjoint covers of the unit interval (or equivalently $\mathbb{R}^n$ for $n\in\mathbb{N}\setminus\{0\}$) by less than $2^\omega$ closed sets in the context of covers of infinite dimensional Banach spaces $\ell_1(\kappa)$ for infinite $\kappa$ under geometric conditions arising in tiling theory and in approximation theory. Specifically, for $\kappa=\omega_1, \omega_2$, using side-by-side Sacks forcing we prove the consistency of an arbitrarily large continuum above $\kappa$ with the existence of a highly disconnected proximinal set in $\ell_1(\kappa)$, built from norm-compact pieces separated by a common positive distance, and the existence of a normal disjoint tiling of $\ell_1(\kappa)$ by star-shaped bodies of the form $X+B$, where $X$ is norm compact and $B$ is the unit ball. We also prove that if $\kappa$ is an uncountable cardinal of countable cofinality, then $\ell_1(\kappa)$ does not admit any normal disjoint tiling by bodies of the form $X+B$, where $X$ is closed and norm-separable and $B$ is the unit ball. These results complement a result of Klee of 1981 obtained for $\kappa$ satisfying $\kappa^\omega=\kappa$, recent results of De Bernardi, Russo, Sezgek and Somaglia, and a $\mathsf{ZFC}$ a machine-discovered result (included in the appendix) that there are no nontrivial discrete Chebyshev sets in $\ell_1(\kappa)$ when $\kappa<2^\omega$.

Let X be a Tychonoff space and B_1(X) be the space of real-valued Baire 1 functions. Using the classical Lebesgue's characterization of Baire 1 functions and the strong Choquet game we prove that if X is locally compact or a topological sum of sigma-compact spaces, the space B_1(X) equipped with the topology \tau_UC of uniform convergence on compacta, is a strong Choquet space. For the Sorgenfrey line S, the space (B_1(S); \tau_UC) is of the first Baire category in itself.

Given a metric space $Y$ and a compact subspace $X$, we present a method for constructing an inverse sequence of finite topological spaces by sampling points from neighborhoods of $X$ in $Y$. We show that these inverse sequences can be employed to define shape theory and to approximate shape invariants, such as Čech homology groups.

We continue the study of selection principles in the uncountable setting initiated in [18] and further developed in [1]. We investigate several problems posed in [1] concerning the Scheepers diagram, the $\kappa$-Hurewicz Conjecture, and related questions on $\kappa$-Menger and $K_\kappa$-spaces. We determine the Scheepers diagram in the uncountable setting up to three unresolved implications and compute the critical cardinalities of several properties occurring in the diagram. For weakly compact $\kappa$, we show that there is a model of ZFC in which the $\kappa$-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at $\kappa$. Moreover, any model in which $\kappa$ is weakly compact and the $\kappa$-Hurewicz Conjecture holds must satisfy $2^{\mathfrak{b}_\kappa} = 2^\kappa$. We also study the existence of $\mathfrak{b}_\kappa$-scale sets which are $K_\kappa$-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no $\mathfrak{b}_\kappa$-scale set is a $K_\kappa$-space. Consequently, under this assumption, every $\mathfrak{b}_\kappa$-scale set yields an alternative example of a $\kappa$-Menger space which is not a $K_\kappa$-space.

A submetric space is a topological space equipped with a continuous metric that generates a metric topology weaker than the original one (e.g., a separable Hilbert space with the weak topology). In this paper, we systematically investigate the compactness-related properties of submetric spaces, which make this class of topological spaces a particularly suitable tool for probabilistic applications. As an illustration, we discuss in detail the compactness of the closure of a càdlàg trajectory taking values in a submetric space.

An $n$-mean on a topological space $X$ is a continuous symmetric map $p\colon X^n\to X$ satisfying $p(x,\ldots,x)=x$ for every $x\in X$. If $X$ is a $G$-space, such an $n$-mean is equivariant if $p(gx_1,\ldots,gx_n)=g p(x_1,\ldots,x_n)$ for every $g\in G$ and $x_1,\ldots,x_n\in X$. For finite $G$, we prove that, in the presence of an equivariant $n$-mean with $n$ a multiple of $\lvert G\rvert$, the $\mathrm{ANE}$, $\mathrm{AE}$, and $\mathrm{AR}$ properties imply their equivariant counterparts. We next consider $\mathbb Z_2$-actions induced by decreasing involutions on topological lattices. We first prove that every fixed point of such an involution on a modular lattice yields an explicit equivariant $2$-mean. Furthermore, we introduce a lattice-theoretic version of the Babylonian iteration used to approximate the geometric mean and prove that, under suitable conditions on the order and its interaction with the topology, this iteration produces an equivariant $2$-mean whenever the lattice admits a $2$-mean compatible with the order. Finally, we apply these results to spaces of compact convex bodies and supercoercive convex functions, where we prove the existence of equivariant $2$-means and show that both spaces are $\mathbb Z_2$-absolute retracts.

In this paper, we study several properties concerning cut points and non-cut points. First, we show that the property of being a continuum consisting entirely of shore points is preserved under refinable maps. Next, we show that having a block point, having a non-shore point, and having a strong center point are not Whitney-reversible properties. Consequently, \cite[Question 2.9]{nonweak} has a negative answer. Furthermore, we show that none of the classes of continua witnessing the failure of the sequential strong Whitney-reversible property for these three properties admits a common model. Finally, in connection with the fact that colocal connectedness is not a Whitney-reversible property, we show that the class of nonaposyndetic continua having a cut point and such that every positive Whitney level is colocally connected has no common model.

In the setting of the previously reported degree-seven counterexample of ani, we present an explicit monic polynomial with real rational coefficients and a direct proof by certified separators. The polynomial has seven simple zeros in the open unit disk. Every continuous path joining distinct zeros in its closed unit sublevel set has image of one-dimensional Hausdorff measure greater than $2+10^{-8}$. Five piecewise-linear graphs exclude twenty of the twenty-one zero pairs; two short forbidden segments force a quantitative detour for the remaining pair. Exact rational Bernstein certificates verify the graph inequalities on forty-four terminal intervals and their unbounded tails. The proof does not require critical-point counting or analytic inverse branches. We separately record the general continuum and dilation arguments that transfer path-length obstructions to Hausdorff measure and to closed sublevel sets. Those transfers are not asserted as new principles. The focus is the explicit real-coefficient construction, its separation certificates, and its verified numerical margin, rather than a first negative answer to the length-two question or a disproof of every larger universal bound.

Let $\mathcal I$ be an admissible ideal on $\mathbb N$. Zhou, Liu, Liu, and Lin asked whether every regular space with a $\sigma$-hereditarily closure-preserving $\mathcal I$-$sn$-network is $\mathcal I$-$sn$-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a local countability theorem. If $y_n\to_{\mathcal I}x$ and $\{n\mid y_n\ne x\}\in\mathcal I^+$, then every point-discrete family of $\mathcal I$-sequential neighborhoods of $x$ is countable. In a $T_1$ space, this applies in particular to every HCP family. If no $\mathcal I$-convergent sequence to $x$ has $\mathcal I$-positive support away from $x$, then $\{x\}$ is an $\mathcal I$-sequential neighborhood. It follows that every space with a $\sigma$-point-discrete $\mathcal I$-$sn$-network is $\mathcal I$-$snf$-countable. This conclusion, together with the characterization theorems of Ge and of Zhou, Liu, Liu, and Lin, settles Problem 5.2. The countability bound cannot in general be strengthened to finiteness. A countable Rudin–Keisler antichain of free ultrafilters gives an admissible ideal for which local HCP-finiteness fails. Every $P^{-}$ ideal has local HCP-finiteness. Preservation theorems for Fubini products, heterogeneous Fubini sums under cross-row Katětov absorption, and suitable increasing unions extend local HCP-finiteness beyond the $P^{-}$ class. They apply to the finite Fubini powers of $\mathrm{Fin}$, Katětov's first limit-stage ideal $\mathrm{Fin}^{\omega}$ and its finite successors, the tree-derived rank-$\omega$ ideal $\mathcal H_{<\omega}$, and the inductive-limit ideal $\mathrm{Fin}_{\omega}$. Every listed example other than $\mathrm{Fin}$ fails $P^{-}$.

We continue the study of the Menger and Rothberger games on lattices carried out in "Pointless proofs of the Menger and Rothberger games" (Topology Appl. 300 (2021), 107774). This time, we extend the earlier results by dropping some hypotheses that turned out to be unnecessary, and use Stone duality to recover known game characterizations for dense open families. We also give a formulation of $\mathsf U_{\mathrm{fin}}$ for partially ordered sets and prove its game characterization without any lattice assumption. Finally, almost disjoint families give complete distributive lattices on which the selection principles and the corresponding games differ. Dias's results give the lower bounds $\operatorname{cov}(\mathcal M)$ and $\mathfrak d$ for the least sizes of such counterexamples.

Lawson and Mislove posed the following problem in 1990 as Problem~535 in Open Problems in Topology: for a core-compact space \(X\) and a dcpo \(P\) equipped with its Scott topology, under what conditions on \(P\) do the Isbell and Scott topologies on \(C(X,P)\) agree?It was proved that, for a nonempty bicomplete continuous dcpo \(P\), the Isbell and Scott topologies on \(C(X,P)\) coincide for every core-compact space \(X\) if and only if \(P\) is bounded complete; for every compact core-compact space \(X\) if and only if \(P\) is conditionally bounded complete; and for every RW-space \(X\) if and only if \(P\) is a pointed continuous \(L\)-domain.We remove the bicompleteness assumption from all three classifications by combining the forbidden-retract theorem of Jia, Jung and Li with a separation theorem for powers of downward well-ordered chains and suitable Alexandrov test spaces.

For every positive sequence that tends to zero faster than every fixed exponential, we construct a Hausdorff topological Vector Group topology on the additive group of real numbers. It lies strictly between the minimal Hausdorff topological Vector Group topology and the usual topology. The construction is illustrated by factorial powers, quadratic exponential decay, and prime radicals divided by a quadratic exponential. We also record a finite scalar covering criterion for comparing two such topologies.

Magnitude is an isometric invariant of metric spaces introduced by Leinster in 2011. It is nowhere continuous on the Gromov–Hausdorff space of finite metric spaces, however, positive continuity results do exist if we restrict the ambient space. In this paper, we prove that magnitude is continuous at every finite subset $F$ of $\ell_1^N$. We do this by first deriving the weight measure for a finite union of cubes, i.e.\ cubical thickenings of the points in $F$, and then showing that these thickenings converge to the magnitude of the underlying finite set.

We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\T^{\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \cite{LMT}.

We prove various properties of the universal valued Abelian groups $\mathbb{G}_r(N)$ constructed by Niemiec, for $r\in\{1,\infty\}$ and $N\in\{0,2,3,\ldots\}$. First we show that the completion of $\Gamma_0=\bigoplus_{n\in\mathbb{N}}\mathbb{Q}/\mathbb{Z}$ (for $N=0$) or $\Gamma_N=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}/N\mathbb{Z}$ (for $N\geq 2$) with respect to a generic invariant metric, bounded by $1$ when $r=1$, is isometrically group-isomorphic to $\mathbb{G}_r(N)$, recovering a result of Doucha in the case when $r=\infty $ and $N=0$. This confirms an expectation of Doucha for the group $\Gamma_N$. We combine this genericity result with a criterion of Melleray and Tsankov concerning the extreme amenability of the generic completion of a countable group to obtain the extreme amenability of $\mathbb{G}_r(N)$. We then establish the strictly stronger properties that these groups are Lévy and strongly exotic. We conclude by showing that $\mathbb{G}_r(0)$ is also monothetic, from which we obtain the existence of a monothetic group structure on the Urysohn sphere, giving a bounded version of a result by Cameron and Vershik and answering a question of Niemiec.

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

We introduce several Ramsey-theoretic properties of 0-dimensional ambits and obtain their dynamical characterizations. In consequence, we obtain Ramsey-theoretic criteria for triviality and for profiniteness of some important invariants (in particular, of the Ellis groups) of the ambits in question. This yields a criterion for the structural property that each distal minimal factor of the given ambit is a profinite flow. Another result is a criterion for metrizability of the minimal left ideals in the Ellis semigroup. Then we study three specializations of the above abstract context: to first order theories, to definable groups, and to the classical Kechris-Pestov-Todorčević theory, recovering known and obtaining new results in each of these contexts.

The unitary group of a von Neumann algebra on an $\aleph_0$-dimensional Hilbert space is polynomially weak$^*$-dense (equivalently, $G_{delta}$ dense or residual) in the normal unit ball if and only if the canonical finite atomic quotient of the von Neumann algebra is trivial. The analogous result holds for the usual weak$^*$ topology, in which case one can drop the normality constraint. In the latter form, this answers a question of T. Eisner and generalizes the counterpart for full bounded-operator algebras.