6,012 papers in this slice of arXiv.
David V. Feldman, Alexander Wilce
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 ω
Anaté Kodjovi Lakmon, Yaogan Mensah
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.
Zhongqiang Yang, Nada Wu
For a T1-space X, let Cld(X) denote all its nonempty closed subsets and T4(X)={(F1,F2)∈Cld(X)2:F1∩F2=∅}. In terms of closed sets, X is T4 iff for each (F1,F2)∈T4(X), there is a pair of closed sets φ1(F1,F2) and φ2(F1,F2) such that their union is X and Fj∩φj(F1,F2)=∅ for j=1,2. Thus, in this paper, for a topology τ on Cld(X), we introduce the definition: A T1-space X is called CT4 for the topology τ if the above maps φj:T4(X)→Cld(X) are continuous on τ. Similarly, for i=1,2,3, we can define a T1-space to be CTi for τ. We only consider the Vietoris topology on Cld(X) and show that every CT4-space is countably compact, and every CT3-space is a Fréchet-Urysohn space, every separable subspace of a CT3-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 CT4. All infinite discrete spaces, all finite-dimensional Euclidean spaces and all countable limit ordinal spaces are CT3 but not CT4. Also, each metrizable space with a unique non-isolated point is CT3, and is CT4 if it is compact. Moreover, the infinite sum of CT3 spaces is CT3 but not CT4. Every countable space with a unique non-isolated point is CT2, and it is CT3 if and only if it is metrizable. All subfields of real numbers and their complement spaces are CT2 but not CT4; and their CT3 status remains unclear. All uncountable ordinal spaces are not CT2. The one-point compactification of any uncountable discrete space is not CT2. CT1 and T1 are equivalent, hence all spaces above are CT1. Open Problems: is there a non-metrizable CT3 or CT4 space? Is any compact CT2 space CT3 or CT4?
Li-Hong Xie, Jiang Yang
We prove that every countable Hausdorff functionally balanced topological group is balanced, answering Question 13 in the problem list of Bouziad and Troallic. The proof does not assume metrizability, first countability, sequentiality, or a non-Archimedean local base. Its non-locally-precompact part combines an infinite-fibre amplification, a simultaneous Bernoulli colouring lemma for countably many arbitrary self-maps, a McShane extension with respect to a continuous left-invariant pseudometric, and the Protasov--Saryev criterion. We also prove a countable-union form of Question 4: in an group, every left uniformly discrete family whose union is countable is right uniformly discrete. Consequently, Questions 7 and 8 have positive answers for ℵ0-bounded groups. We reconstruct all thirteen questions and their logical relations. The original problem chapter states, without proof, that a positive answer to Question 13 would imply Question 4 for all ℵ0-bounded groups. We isolate the ambient reflection principle needed for that transfer and explain why neither subgroup reduction nor the present colouring proof establishes it. Therefore Question 12 is not claimed as solved here: the precise boundary between the proved countable theorem and the asserted ℵ0-bounded transfer is recorded explicitly.
Ľubica Holá
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 τ_p be the topology of the pointwise convergence. For (Q(X), τ_p) complete metrizability, Polishness and Cech-completeness are equivalent. If (Q(X), τ_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), τ_p) is completely metrizable if and only if X is countable and I(X) is dense in X.
Xiaoyong Xi, Chong Shen, Dongsheng Zhao
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(Rn) is not sober for every positive integer n. In particular, RO(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.
Tattwamasi Amrutam
A discrete group Γ has the C∗--ISR property if every unital Γ--invariant C∗--subalgebra A⊆Cr∗(Γ)
Mikołaj Krupski, Kacper Kucharski
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 Cp(X) onto the space Cp(Y) and the space X is either strongly σ-scattered or has property (κ), 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.
Mikołaj Krupski, Kacper Kucharski
In this note, we prove that an arbitrary product of W-spaces is a κ-Fréchet-Urysohn space. We also show that the boundary tightness of a product X×Y is at most κ provided that X is compact with tightness t(X)≤κ and Y has boundary tightness tb(Y)≤κ. The first result fully resolves, and the second partially addresses, two questions recently raised by Tkachuk.
Tomoki Uda
For a discrete topological space D, let 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 Met(D) is not completely metrizable when ∣D∣=ℵ1
Vinicius de Oliveira Rodrigues, Paul Jan Szeptycki, Artur Hideyuki Tomita
Countable compactness need not be preserved by finite products. Motivated by compactness notions for maps indexed by finite subsets of ω, we introduce cascade countable compactness for arbitrary barriers. For B=[ω]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≤k<ω, then k-cascade countable compactness of G implies that Gk is countably compact. Cascade countable compactness for the Schreier barrier implies that Gω is countably compact. In contrast, we construct a Tychonoff space that is n-cascade countably compact for every n<ω but has a non-countably compact square, and a Hausdorff Boolean group H that is B-countably compact for every barrier B but whose square is not countably compact. We also construct a Hausdorff Boolean group without nontrivial convergent sequences that is B-cascade countably compact for every barrier B. Finally, we obtain a subspace X⊆βω such that Xκ is n-cascade countably compact for every κ<h and every n<ω, whereas expX is not pseudocompact.
Marco Abbadini
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×[0,m], for every approximate identity and every ε>0, one member of the approximate identity has radius excess (i.e., the output radius minus the input radius) uniformly at most ε. 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×[0,m]. This solution was obtained independently of the recent work of Chen, Kou, and Lyu and uses a different method.
Wei Ji, Xiaoquan Xu
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×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 Σ B is not co-sober, thereby answering negatively a question on Scott spaces of complete Boolean algebras. The same Scott space Σ B is non-sober and, as a Scott space of a complete lattice, is well-filtered.
Evgeny Kuznetsov
For a commutative Bezout ring R, we give a criterion, in terms of colon ideals with principal radical, for the lattice K˚(Spec(R)) of compact open subsets of Spec(R) to be a Heyting algebra. Bezhanishvili and Tressl showed that K˚(Spec(C(T))) is pseudocomplemented whenever T is a basically disconnected compact Hausdorff space, and asked whether Spec(C(βN)) is actually an Esakia space. We show it is not: applying our criterion to C(βN)≅ℓ∞(N,R) produces a diagonal counterexample, and the same obstruction rules out βD for every infinite discrete D. A grid-existence theorem for σ-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 Spec(C(T)) an Esakia space.
Souvik Mandal, Ankur Sarkar
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.
Souvik Mandal
This article investigates the coprimality topology on the set of non-zero Gaussian integers, Z[i]∖{0}, generated by the arithmetic basis σα={β=0:gcd(α,β)∼1}. By analyzing prime supports up to associates, we establish that two points are topologically indistinguishable if and only if their supports coincide. We demonstrate that the space is both hyperconnected and ultraconnected, and we explicitly characterize its Kolmogorov quotient X as the space of finite subsets of associate classes of Gaussian primes, endowed with the basis OF={S∈X:S∩F=∅} for F∈X. Finally, we demonstrate that the set of Gaussian primes is dense in Z[i]∖{0} with respect to the coprimality topology, thereby establishing a topological proof of the infinitude of Gaussian primes.
João Areias, Jorge Picado
Following a suggestion in a paper by Erné, Picado and Pultr, we establish the category of basic zero-dimensional spaces and their basic continuous maps. The objects are closure spaces with a distributive closure system containing a specified meet-base of complemented members that make the category a common generalization of those of closure spaces, zero-dimensional topological spaces, Heyting semilattices, and locales (frames). We treat images, preimages, continuity, openness and closedness of maps between basic zero-dimensional spaces. Among our main results, we have a useful description of preimages for basic continuous maps (analogous to the one for localic preimage maps as coframe homomorphisms), and a Joyal-Tierney type theorem for basic open maps. A novel aspect of the category of basic zero-dimensional spaces is a duality principle that, among other things, enables results for basic closed maps to be obtained for free from those for basic open maps.
Soumajit Dey, Sudip Kumar Acharyya, and Dhananjoy Mandal
Let X be a topological space. For each transfinite cardinal number ℵα, we define a topology Cℵα(X) on the ring C(X). With ℵα=ℵ0, Cℵα(X) reduces to the space Cp(X). We prove that for ℵα≥ℵ1, Cℵα(X) is pathwise connected if and only if it is connected if and only if X is pseudocompact. Here, we define ℵα-separable space and furthermore, we show that X is ℵα-separable when and only when Cℵα(X) is metrizable when and only when it is a sequential space. Later we introduce two new cardinal functions, namely ccℵα(X) and acℵα(X) which turned out to be the character and pseudocharacter of Cℵα(X) respectively. At the end of this article we show that a number cardinal functions associated with the space Cℵα(X) are equal.
Khemraj Shukla, George Em Karniadakis
Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space (V,{pα}α∈A), whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative L2 error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual V′, including for non-normable input spaces.
Yaé U. Gaba
We extend the -fixed-point and -startpoint framework for quasi-uniform spaces. We reformulate the original existence framework through the filter axioms and supply a two-sided contraction condition on a generating family of quasi-pseudometrics. For this condition we prove a Banach-style fixed-point theorem on T0 bicomplete quasi-uniform spaces, with a Picard-type error bound; a Boyd--Wong companion and a stability estimate follow. The conjugate quasi-uniformity yields an endpoint version, a selector argument extends the result to multivalued maps and to common startpoints of commuting families, a Knaster--Tarski variant covers monotone selectors on order-complete spaces, and a direct quasi-Hausdorff route handles weakly contractive multivalued maps. Moreover, the T0-quotient removes the T0 assumption from the corresponding involution-based fixed-point theorem.