Papers

We study how the noncentral zeros of an elliptic-curve $L$-function enter the Birch–Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed $L$-function yields an exact Hadamard decomposition, and the zero statistics $J_E$ and $V_E$ satisfy the positive identity $J_E=V_E/8+\Delta_E$. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average $82.4%$ of $\Delta_{E,20}$, while independent edge values reveal a discrete $35/36$ zero-count split beyond height $20$. For Haar $SO(2N+1)$ we prove a fixed-dimensional hard-edge law $\Pr{J_N>y}\sim C_Ne^{-3y/2}$. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.

We introduce Menger and Rothberger selection principles and games for convergence spaces. Alice plays families that meet every convergent filter, and Bob's selections are required either to retain this property or merely to cover the underlying set. When the convergence is topological, both games recover the classical games. The winning condition requiring an $L$-cover satisfies analogues of the Hurewicz and Pawlikowski characterizations, the condition requiring a cover of $X$ is represented by the weak Menger and Rothberger games. We also show that, for a regular convergence space, a winning strategy for Bob in $\mathsf G_{\fin}(\mathcal C_L,\Cov(X))$ implies an Alster-type covering property. Under hereditary Lindelöfness the space is moreover a countable union of compactoid subsets, which are compact in the pretopological case.

In this paper we prove the equality of the dimensions IndX and dimX for any stratifiable spaces. As a consequence, we obtain the equality of the dimensions IndX, indX and dimX for stratifiable spaces with a countable network. It has been proven that every stratifiable space is an S-space (possesses an S-network).

Let X be a metrizable space, and let Met(X) denote the space of metrics compatible with the topology of X, regarded as a subspace of the space of continuous pseudometrics with the supremum-metric topology. We first prove that if Z is a discrete space of cardinality aleph-one, then Met(Z) is not Borel. As a consequence, if Met(X) is Borel, then X is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space X is completely metrizable if and only if X is sigma-compact. We also establish non-Archimedean analogues for spaces of ultrametrics.

On this article we provide a relation between two inherently different semantic structures for intuitionistic modal logic. We start by recalling the Heyting Algebras, then defining the language and the axioms for the iS4h intuitionistic calculus. We then proceed to analyse two structures discussed on [4] and their relation. Finally, we discuss about a structure defined on [3, Chapter 6], establishing properties in order for it to be sound with iS4h, and finally conclude that it is semantically equivalent to the other two structures defined by [4].

In this paper, we study Levine equivalence in aura topological spaces. We show that the aura topology is determined by the reachability preorder induced by the scope function and is therefore an Alexandrov topology. We prove that the aura-Levine hull has the explicit representation $K_{\mathfrak a}(A)=\uparrow_{\mathfrak a}A=S_{\mathfrak a}^{\infty}(A)$, and hence two subsets are aura-Levine equivalent if and only if they have the same eventual forward spread. We also describe the quotient structure of the equivalence classes, characterize separation properties through the reachability preorder, classify scope functions that induce the same Levine equivalence, and examine preservation under aura-continuous mappings. For finite aura spaces, we provide an explicit procedure for deciding aura-Levine equivalence using the reflexive-transitive closure of the scope relation.

Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h\colon C_p(K)\to E_w$ whose inverse is continuous at $h(0)$. Consequently, $C_p(K)$ and $C_w(L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.

We give a systematic account of the shape theory of $\infty$-topoi, viewing the shape of an $\infty$-topos as its generalized homotopy type. We establish the basic functorial properties of the shape, including preservation of colimits, descent, and homotopy invariance. We then prove that shape preserves cofiltered limits under compactness and perfectness hypotheses and establish Künneth-type formulas for products. Finally, we give conditions under which the shape of an $\infty$-topos determines its cohomology and homology.

We prove in ZFC that every torsion-free Abelian group of cardinality $\mathfrak c$ admits a Hausdorff countably compact group topology without nontrivial convergent sequences. In particular, this applies to the free Abelian group $\mathbb{Z}^{(\mathfrak c)}$, the Baer-Specker group $\mathbb{Z}^\omega$ and $\mathbb{Q}^{(\mathfrak c)}$. For the topology constructed on $\mathbb{Z}^{(\mathfrak c)}$, the coordinatewise nonnegative cone is countably compact in the subspace topology. Consequently, there exists in ZFC a commutative Tychonoff countably compact topological semigroup which has two-sided cancellation but is not a group, giving a negative answer to Wallace's question. Combined with earlier results, the main theorem also yields in ZFC a Tychonoff countably compact topological semigroup containing a copy of the bicyclic semigroup and a functionally Hausdorff countably compact paratopological group that is not a topological group.

Recently, in the study of topological properties in Domain theory, the first countable spaces and Frechet-Urysohn spaces receive close attention and are frequently employed. In this paper, we focus on sequential spaces weaker than Frechet-Urysohn spaces. The main results are: (1) For a well-filtered space X, the Scott topology and the upper vietoris topology on K(X) coincide if PS(X) is a sequential space; (2) every {\omega}-well-filtered coherent d- space X is sober when the procuct of X is a sequential space.

Let $X$ be an ultrametrizable space. We study the space of bounded compatible ultrametrics on $X$, equipped with its natural non-Archimedean distance. For every positive integer level, we prove that additive absolute Borel complexity of this moduli space implies multiplicative absolute Borel complexity of $X$ at the same level, and conversely. We also prove that an ultrametrizable space is a countable union of locally compact subspaces if and only if it is a countable union of closed subsets in every completion induced by a bounded compatible ultrametric. As a consequence, this moduli space is completely metrizable exactly when $X$ is a countable union of compact subsets.

In this short note, we show that some well-known filtrations of infinite dimensional groups of Fredholm operators associated to certain perturbation classes are, in fact, Fredholm $\Delta$-filtrations (see Definition 5 below.) For example, let $\E$ be a separable infinite dimensional real Hilbert space. The group $\GLK(\E)$ of all invertible operators on $\E$ which are compact perturbations of the identity is the structure group for Hilbert Fredholm manifolds and bundles modeled on $\E$ \cite{ElwTr, Ksch, Mkhr}. Using an orthonormal basis, there are canonical inclusions of general linear groups: $$\GL(1) \subset \cdots \subset \GL(n) \subset \GL(n+1) \subset \cdots \subset \GL(\infty) = \varinjlim \GL(n) \subset \GLK(\E).$$ We show this is a Fredholm $\Delta$-filtration of the Fredholm manifold $\GLK(\E)$ with dimension sequence $\Delta(n) = \dim(\GL(n)) = n^2$, which was not discussed in the classical Fredholm manifold literature because of the rigid constraint that the dimensions of a filtration increase only by one.

This article shows that the exact completion as a regular category of the homological category of Polish groups and continuous group homomorphisms is the category of groups with a Polish cover and Borel-definable group homomorphisms. We also obtain several equivalent characterizations for the morphisms in the exact completion. An analogous description is deduced for several important subcategories of the category of Polish groups.

Say that a topological space $X$ has finite, respectively bounded, cliques iff every closed, irreflexive binary relation — equivalently, every closed, loop-free directed graph — on $X$ has cliques of finite, respectively bounded finite, size. Every compact space has bounded cliques. Having finite cliques implies limit-point compactness, and is implied by $\omega$-limit point compactness (equivalently, countable compactness). Thus, for $T_1$ spaces, having finite cliques is equivalent to countable compactness. Having bounded cliques is strictly weaker than compactness. Indeed, any space $X$ such that $X^{\omega}$ is countably compact has bounded cliques. However, we have found no example of a countably compact space having finite but unbounded cliques. The existence of such a space is the major open problem raised in this note.

We prove a Hilbert space-valued analogue of Pego's compactness theorem on compact groups. For square-integrable functions taking values in a Hilbert space rather than in the complex numbers, we show that a bounded family is precompact exactly when it is simultaneously well behaved in two complementary senses: its members do not change much under small translations of the group, and their Fourier coefficients decay uniformly across the family. This equivalence holds without restriction when the Hilbert space is finite-dimensional, and it specializes to the known scalar-valued theorem when the Hilbert space is just the complex numbers. We then construct an explicit example showing that the equivalence genuinely breaks down once the Hilbert space is allowed to be infinite-dimensional. To repair this, we introduce a uniform tightness condition and we show that under this extra hypothesis the equivalence is restored regardless of the dimension of the Hilbert space. Along the way we establish the Plancherel isometry, the Hausdorff-Young inequality and its inverse for this vector-valued Fourier transform.

For a $T_1$-space $X$, let $Cld(X)$ denote all its nonempty closed subsets and $T_4(X)=\{(F_1,F_2)\in Cld(X)^2:F_1\cap F_2=\emptyset\}$. In terms of closed sets, $X$ is $T_4$ iff for each $(F_1,F_2)\in T_4(X)$, there is a pair of closed sets $\phi_1(F_1,F_2)$ and $\phi_2(F_1,F_2)$ such that their union is $X$ and $F_j\cap \phi_j(F_1,F_2)=\emptyset$ for $j=1,2$. Thus, in this paper, for a topology $\tau$ on $Cld(X)$, we introduce the definition: A $T_1$-space $X$ is called $CT_4$ with respect to $\tau$ if the above maps $\phi_j:T_4(X)\to Cld(X)$ are continuous on $\tau$. Similarly, for $i=1,2,3$, we can define a $T_1$-space to be $CT_i$ for $\tau$. We only consider the Vietoris topology on $Cld(X)$ and show that every $CT_4$-space is countably compact, and every $CT_3$-space is a Fréchet-Urysohn space, every separable subspace of a $CT_3$-space is metrizable. We give relevant examples. Any finite-dimensional cubes, the infinite-dimensional cube, any finite-dimensional spheres, and all 0-dimensional compact metrizable spaces are $CT_4$. All infinite discrete spaces, all finite-dimensional Euclidean spaces and all countable limit ordinal spaces are $CT_3$ but not $CT_4$. Also, each metrizable space with a unique non-isolated point is $CT_3$, and is $CT_4$ if it is compact. Moreover, the infinite sum of $CT_3$ spaces is $CT_3$ but not $CT_4$. Every countable space with a unique non-isolated point is $CT_2$, and it is $CT_3$ if and only if it is metrizable. All subfields of real numbers and their complement spaces are $CT_2$ but not $CT_4$; and their $CT_3$ status remains unclear. All uncountable ordinal spaces are not $CT_2$. The one-point compactification of any uncountable discrete space is not $CT_2$. $CT_1$ and $T_1$ are equivalent, hence all spaces above are $CT_1$. Open Problems: is there a non-metrizable $CT_3$ or $CT_4$ space? Is any compact $CT_2$ space $CT_3$ or $CT_4$?

We prove that every countable left uniformly discrete subset of a Hausdorff functionally balanced topological group is right thin. As applications, we answer two questions of Bouziad and Troallic: every left precompact subset of a Hausdorff functionally balanced group is right precompact, and every Hausdorff $\omega$-narrow functionally balanced group is a SIN group.

Quasicontinuous functions have found applications in many areas of mathematics. We study completeness properties of the space of quasicontinuous functions equipped with the topology of pointwise convergence. Let X be a Hausdorff topological space, Q(X) be the space of quasicontinuous real-valued functions and {\tau}_p be the topology of the pointwise convergence. For (Q(X), {\tau}_p) complete metrizability, Polishness and Cech-completeness are equivalent. If (Q(X), {\tau}_p) is completely metrizable, then X is countable and the set I(X) of isolated points of X is dense in X. If X is first countable, then (Q(X), {\tau}_p) is completely metrizable if and only if X is countable and I(X) is dense in X.

For a topological space X, let RO(X) be the complete Boolean algebra of regular open subsets of X, ordered by inclusion. We prove that, for every second-countable T3 space X, the Scott space of RO(X) is sober if and only if the set of all isolated points of X is dense in X. Consequently, RO$(\mathbb R^n)$ is not sober for every positive integer $n$. In particular, RO$(\mathbb R)$ is not sober, which provides an answer to an open problem concerning the sobriety of complete Boolean algebras. This characterization also yields a systematic way to obtain more natural examples of complete lattices whose Scott spaces are non-sober.

We study maximal (proper) filters in the complete lattice of partitions of an infinite set~$X$. In the language of uniform spaces, these are precisely the atoms of the lattice of zero-dimensional uniformities on $X$, introduced by Pelant and Reiterman and studied further by Pelant, Reiterman, Rödl and Simon. The first half of this paper recovers, sharpens, and extends their classification in purely partition-theoretic terms. {Call a maximal filter of partitions type I if it does not contain all finite partitions, and type II if it does.} Type I filters are induced, in an essentially unique way, by ultrafilters on families of pairwise disjoint doubletons. {A type II filter} determines a non-principal ultrafilter on $X$, the heart; the heart determines the filter precisely when {the former} is minimal in the Rudin–Keisler order. Each member $F$ of {a type II filter} gives rise to a closed fiber in $X^{*} = \beta X \setminus X$, {consisting of} the set of ultrafilters agreeing with the heart on $F$. We prove a trichotomy describing the topology of arbitrary fibers. We then show that the fibers do not encode the filter: fibers do not form a semilattice under intersection, the closure of an infinite discrete set of ultrafilters of a single Rudin–Keisler type (a sparse set, in our terminology) need not be a fiber, and a partition incompatible with a member of the filter may have a {fiber strictly larger than that member.} A representation that does succeed is nevertheless available in another category: the {type-II filters} with heart $\mathfrak{u}$ correspond to the maximal proper substructures of the ultrapower $X^X\!/\mathfrak{u}$ of the full structure on $X$. The topological representation problem remains open.

We give negative answers to two questions of X.Xu concerning the occurrence of the unit interval in nonquasialgebraic quasicontinuous domains. We construct, directly from the binary tree and the binary-value map, a well-founded quasicontinuous dcpo \[ P=2^{<\omega}\mathbin{\dot\cup}[0,1] \] which is not quasialgebraic. Moreover, $P$ contains no sub-dcpo isomorphic to $[0,1]$, and $[0,1]$ is not a Scott-continuous retract of $P$.

A discrete group $\Gamma$ has the \emph{$C^*$–ISR property} if every unital $\Gamma$–invariant $C^*$–subalgebra $A\subseteq C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleft\Gamma$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$–ISR property for every $n\geq3$.

We prove that if $X$ and $Y$ are first-countable perfect spaces such that the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is a hereditarily Baire space if and only if $Y$ is hereditarily Baire as well. We also establish that for any Tychonoff space, if there exists a continuous linear surjection of the space $C_p(X)$ onto the space $C_p(Y)$ and the space $X$ is either strongly $\sigma$-scattered or has property $(\kappa)$, then $Y$ also satisfies those properties. Additionally, we obtain the following result: if $X$ and $Y$ are Tychonoff spaces in which every closed set has a $W$-point, and if the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is scattered if and only if $Y$ is scattered.

In this note, we prove that an arbitrary product of $W$-spaces is a $\kappa$-Fréchet-Urysohn space. We also show that the boundary tightness of a product $X \times Y$ is at most $\kappa$ provided that $X$ is compact with tightness $t(X) \leq \kappa$ and $Y$ has boundary tightness $tb(Y) \leq \kappa$. The first result fully resolves, and the second partially addresses, two questions recently raised by Tkachuk.

For a discrete topological space $D$, let $\mathrm{Met}(D)$ denote the set of metrics on $D$ that are compatible with the discrete topology, equipped with the topology induced by the supremum distance. Ishiki's Conjecture 5.1 asserts that $\mathrm{Met}(D)$ is not completely metrizable when $\lvert D\rvert = \aleph_1$. We prove this in ZFC by constructing a set $A \subseteq [0,1]$ of cardinality $\aleph_1$ that is not $F_\sigma$ and embedding its complement as a closed subspace of $\mathrm{Met}(D)$. The proof has also been formalised in Lean 4.

Countable compactness need not be preserved by finite products. Motivated by compactness notions for maps indexed by finite subsets of $\omega$, we introduce cascade countable compactness for arbitrary barriers. For $\mathcal B=[\omega]^2$, this is the previously studied notion of being doubly countably compact. We show that, in ZFC, if $G$ is a Hausdorff topological group and $1\leq k<\omega$, then $k$-cascade countable compactness of $G$ implies that $G^k$ is countably compact. Cascade countable compactness for the Schreier barrier implies that $G^\omega$ is countably compact. In contrast, we construct a Tychonoff space that is $n$-cascade countably compact for every $n<\omega$ but has a non-countably compact square, and a Hausdorff Boolean group $H$ that is $\mathcal B$-countably compact for every barrier $\mathcal B$ but whose square is not countably compact. We also construct a Hausdorff Boolean group without nontrivial convergent sequences that is $\mathcal B$-cascade countably compact for every barrier $\mathcal B$. Finally, we obtain a subspace $X\subseteq\beta\omega$ such that $X^\kappa$ is $n$-cascade countably compact for every $\kappa<\mathfrak h$ and every $n<\omega$, whereas $\exp X$ is not pseudocompact.

Every RB-domain is an FS-domain. Whether the converse holds was a long-standing open problem. We prove that the domain of closed axis-parallel squares in the plane whose centres lie in the unit square $[0,1]^2$, with the whole plane adjoined and ordered by reverse inclusion, is an FS-domain but not an RB-domain. The proof is quantitative. A finite-grid argument first shows that, on every finite slab $[0,1]^2\times[0,m]$, for every approximate identity and every $\varepsilon>0$, one member of the approximate identity has radius excess (i.e., the output radius minus the input radius) uniformly at most $\varepsilon$. By contrast, for every deflation and every $m>0$, the radius excess is at least $m/(4m+1)$ at some point of the slab $[0,1]^2\times[0,m]$. This solution was obtained independently of the recent work of Chen, Kou, and Lyu and uses a different method.

In this paper, we first prove that for a complete Boolean algebra $L$, the complement graph of $L$ is a KC-space (as a subspace) and each non-singleton compact irreducible subspace of that graph generates a non-principal $k$-irreducible compact saturated set in the Scott space of the square algebra $L\times L$. We then show that every compact sequential US-space embeds into the complement graph of a suitable complete Boolean algebra. The embedding is built from a finite tail-constraint poset and its regular-open completion. Applying the construction to van Douwen's compact Fréchet anti-Hausdorff US-space gives a complete Boolean algebra $B$ whose Scott space $\Sigma~\!\!B$ is not co-sober, thereby answering negatively a question on Scott spaces of complete Boolean algebras. The same Scott space $\Sigma~\!\!B$ is non-sober and, as a Scott space of a complete lattice, is well-filtered.

For a commutative Bezout ring R, we give a criterion, in terms of colon ideals with principal radical, for the lattice $\mathring{\mathcal{K}}(\mathrm{Spec}(R))$ of compact open subsets of $\mathrm{Spec}(R)$ to be a Heyting algebra. Bezhanishvili and Tressl showed that $\mathring{\mathcal{K}}(\mathrm{Spec}(C(T)))$ is pseudocomplemented whenever T is a basically disconnected compact Hausdorff space, and asked whether $\mathrm{Spec}(C(\beta\mathbb{N}))$ is actually an Esakia space. We show it is not: applying our criterion to $C(\beta\mathbb{N}) \cong \ell^\infty(\mathbb{N},\mathbb{R})$ produces a diagonal counterexample, and the same obstruction rules out $\beta D$ for every infinite discrete D. A grid-existence theorem for $\sigma$-complete Boolean algebras lets us push the construction to every basically disconnected compact Hausdorff space, settling the Bezhanishvili-Tressl question completely: for no infinite compact Hausdorff space T is $\mathrm{Spec}(C(T))$ an Esakia space.

We develop a foundational theory of soft quotient topology, providing a systematic approach to quotient constructions in soft topological spaces. We establish the universal property of soft quotient topology and investigate the relationship between global soft topology and its parametric slices, noting that slice-wise quotient behaviour is not sufficient to characterise soft quotients. The usefulness of the framework is illustrated through the construction of soft quotient spaces, together with a study of soft group actions and their orbit spaces. We propose a precise definition of soft covering maps that resolves inconsistencies found in the existing literature. Finally, to illustrate the applicability of our framework, we discuss a potential application in multi-agent motion planning, showing how it can significantly reduce combinatorial complexity under parametric uncertainty.