A proof that dim = Ind for every stratifiable space
The proof builds an intrinsically special network that forces dim and Ind to agree for every stratifiable space.
math.GN
General Topology
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties
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Regular-open algebras: sobriety equals isolated-point density
The result settles an open question: RO(R) has a non-sober Scott space.
· “Sobriety of Regular Open Algebras in Second-Countable T3 Spaces”
All invariant C*-subalgebras of PSL_n(Z) arise from normal subgroups
For n≥3, PSL_n(Z)'s reduced C*-algebra has no exotic invariant subalgebras — rigidity matches the von Neumann case.
Log-minor coordinates certify proper matrix actions
Two explicit coordinate-map tests decide when a subgroup acts properly on an upper-triangular homogeneous space.
· “Proper Actions on Homogeneous Spaces of the Upper Triangular Matrix Group via Coarse Geometry”
Ideal is analytic P iff all its circle subgroups are Polishable
The equivalence is witnessed by an explicit metric, and a dichotomy splits the subgroups into countable or Fσδ-complete.
· “Complexity and Polishability of characterized subgroups on the unit circle”
Two-element chain defines maximal faithful class of dcpos
Inside the class, isomorphic Scott-closed lattices force isomorphic dcpos; any larger class breaks the invariant
Convergence covering games reduce to classical filter-space games
Representation theorem transfers Menger and Rothberger characterizations to convergence spaces.
Reachability preorder decides Levine equivalence in aura spaces
Levine-equivalent sets are exactly those sharing the same eventual forward spread under the scope function.
Finite cliques equal countable compactness
In T1 spaces. Bounded cliques sit strictly below compactness, and the gap between them is unresolved.
Uniform tightness restores Pego compactness in infinite dimensions
The scalar Pego theorem is recovered when the value space is the complex numbers.
· “Pego theorem for Hilbert space-valued functions on compact groups”
Countable discrete sets are right thin in FSIN groups
Two open questions on precompactness and omega-narrow groups are answered affirmatively.
· “Countable uniformly discrete sets in functionally balanced groups”
Complete quasicontinuous spaces: base countable
Complete metric exists exactly for countable domains with dense isolated points.
· “Completeness properties of the space of quasicontinuous functions”
Hereditary Baire property is A-invariant
If free Abelian groups of two first-countable perfect spaces match, both are hereditarily Baire or neither is.
· “On the A-invariance of the hereditary Baire property and related results”
Every product of W-spaces has open-set sequence convergence
Settles the product question: closure points of open sets in W-space products are reached by sequences.
· “The Cartesian product of any family of W-spaces is kappa-Fr\'{e}chet-Urysohn”
Aleph-one discrete: Met(D) has no complete metric
A closed embedding of a non-G_delta set into Met(D) settles the conjecture in ZFC.
· “On Ishiki's Conjecture: Met(D) Is Not Completely Metrizable for lvert Drvert=aleph₁”
Cascade compactness forces countable compactness in group powers
A recursive ultrafilter-limit property transfers from a group to its finite or countable powers, in ZFC alone.
Formal squares are FS-domains but not RB-domains
A concrete square-based domain ends the search for an FS-domain that is not a retract of a bifinite one
· “Formal squares over the unit square separate FS-domains from RB-domains”
Scott spaces of complete Boolean algebras can fail co-sobriety
Construction yields a well-filtered Scott space that is neither sober nor co-sober, settling a 2026 question.
No infinite compact T makes Spec(C(T)) an Esakia space
A diagonal counterexample plus a grid theorem closes the open question for all infinite compact Hausdorff spaces.
Soft quotients and coverings get a precise theory and an N! speedup
A parameter-dependent soft topology lets identical agents plan in a quotient space N! times smaller, with local structure preserved.
· “A Precise Treatment of Soft Quotient Topology and Soft Covering Maps”
A coprimality topology proves infinitely many Gaussian primes
Prime-support equality defines the quotient, and density of Gaussian primes forces their infinitude.
· “A Coprimality Topology on the Gaussian Integers: Kolmogorov Quotient and Gaussian Prime Density”
One category unifies four notions of space
Every basic continuous map gets a preimage that is a coframe homomorphism; open maps obey a Joyal-Tierney theorem.
· “Basic zero-dimensional spaces: a unifying framework for continuity and openness”
New topology on C(X): connected iff X is pseudocompact
For every cardinal above countably infinite, pathwise connected, connected, and the ring property reduce to pseudocompactness.
Functional coordinates make DeepONets discretization-portable
Adaptive topological inputs hit 1.685% Navier-Stokes error with 128 dual coordinates and keep 5.5% Darcy error on unseen grids.
Asymmetric spaces get a two-sided Banach fixed-point theorem
In spaces where closeness is one-way, Picard iteration's forward and backward errors keep their own geometric rates.
· “On U-startpoints and U-fixed points in quasi-uniform spaces: a revisit and extensions”
Subfitness and fitness become closure-operator equations
The same operators that detect subfitness and fitness in locales work on any coframe, with new supplement formulas.
· “Fitness and subfitness via interior and closure operators”
Cocountable real line fits a bounded-complete dcpo model
Construction from almost-disjoint sets settles the open case and yields a non-sober Scott space.
· “The set of real numbers with the cocountable topology has a bounded complete dcpo model”
Co-soberness fails under the Smyth power construction
A T1 co-sober space built on a compact anti-Hausdorff base answers an open problem negatively.
· “Smyth power space of a co-sober space is not always co-sober”
Finite products preserve SI-compactness
Two open questions settled: the product result holds, while discrete naturals have no SI-compact reflection.
Under CH, dense-set choice splits Katětov classes
Two countable dense sets in an uncountable product can yield non-equivalent coordinate ideals, so D is not a harmless choice.
· “Dense-set dependence in the Katv{e}tov order for uncountable coordinate ideals”
There exists a countable compact space that is not SI-compact
It settles the open question and yields a new Scott-non-sober frame.
· “A new method for constructing a compact and non SI-compact space”
The real line's regular-open Boolean algebra is not sober
The diagonal of disjoint regular opens is an irreducible Scott-closed set with no point.
· “The complete Boolean algebra of regular open sets in real line is not sober”
Opposite of compact Hausdorff spaces is not finitely concrete
A topological proof shows every faithful set-valued functor misses one sequential colimit — the chain of finite cubes.
· “A topological proof that compact Hausdorff spaces are not finitely co-concrete”
Exotic surfaces lose ambient symmetries one at a time
A new construction produces topologically identical surfaces whose smooth symmetries shrink stepwise until none remain.
· “Exotic knottings and symmetries of surfaces in 4-manifolds”
d-spectral spaces are exactly the spectra of d-lattices
A two-topology version of spectral-space duality, with patch and de Groot dual properties intact.
Finite-label uniformization = Δ-property on ladder-system spaces
For every carrier, uniformization, the Δ-property, and countable metacompactness coincide.
· “Carrier ideals, tail obstructions, and remainder traces for ladder-system spaces”
Quasi-liminf convergence is topological exactly on WLH-spaces
It pins down when a net's quasi-limit is a topological limit, recovering Lawson convergence for quasicontinuous posets.
· “Characterization of T₀-spaces for quasi-liminf convergence being topological”
One-sided FS closure fails: continuity not preserved
A constructed pair of domains yields a non-continuous Scott function space; the tie domain shows where continuity still survives.
Convergence Choquet-complete T1 spaces get domain models
A singleton winning strategy for the strong Choquet game already builds the model, so sobriety is not needed.
· “Convergence Choquet-complete spaces and domain representations”
Two open dcpo-model problems answered by counterexamples
A well-filtered compact space escapes one model class; a non-k-space still gets a bounded complete model.
Raikov remainders pass to quotients when the kernel is complete
For Raikov-complete almost metrizable kernels, the quotient's remainder is exactly the image of the original remainder.
· “Raikov Remainders of Quotients by Raikov-Complete Almost Metrizable Normal Subgroups”
Dense approximate periods force a zero-one law for sets
In Banach spaces, a Baire set with a somewhere-dense family of approximate periods must be meager or comeager.
Every radical on a homological category yields a factorization system
Radicals familiar from module torsion theory now determine how every map factors through a torsion-free part.
· “The factorization system of a radical on a homological category”
Pseudocompactness survives bounded uniform homeomorphisms
For C_p^* function spaces, a bounded uniform homeomorphism forces the codomain to be pseudocompact if the domain is.
· “Bounded uniform homeomorphisms between C_p^*-spaces preserve pseudocompactness”
Compact subspaces of normal spaces are effective codescent
Pushouts along such inclusions stay Hausdorff, so the inclusion has the exactness property called effective codescent.
· “Effective codescent morphisms of Hausdorff topological spaces”
Hausdorff spaces are exactly those no rough family can rebuild
Rough open sets build coarser topologies from discrete ones, resting on one net-convergence assumption.
· “Unlocking novel topological structures via rough families”
Directional entropy profile separates shapes a single entropy cannot
The Persistent Entropy Transform packs geometry into 64-number vectors that classify ECGs at near-raw-signal accuracy.
· “Persistent Entropy Transform: An entropy-based descriptor for topological data analysis”
Stable maps are exactly the H-covering maps
Abstract completeness-preserving maps now have a concrete description in terms of countable compact sets.
A T0-space is strong R exactly when its closed sets are dual-compact
This compactness test yields tau = tau^dd for locally compact spaces and settles heredity and reflectivity questions.
Digital cycles form fields but resist topological fields
Prime-length simple k-cycles are isomorphic to Z_l, yet their multiplication breaks digital continuity.
Cocompact subgroups inherit their Polish group's coarse geometry
Coarse proper actions and Dirichlet domains show the subgroup's intrinsic geometry is the whole group's.
Bounded G-orbit equals invariant norm
Locally algebraic p-adic representations: a bounded orbit is exactly an invariant norm, built as the pointwise supremum.
Two continuous pieces suffice for the simplest discontinuous class
Two pieces now, not three, plus a local test for which discontinuity level a function sits at
Risk tiers from gene-expression shape survive held-out test
Patients grouped by a pure topology network keep their survival ordering after adjusting for age, stage, and treatment.
· “Topology-Informed Survival Analysis of Breast Cancer Patients Using the Mapper Algorithm”
Lower-curvature Alexandrov geometry from comparison to doubling
Globalization, splitting, volume comparison and the two-dimensional precursor, in one introduction
· “Lectures on Alexandrov spaces with curvature bounded below”
Levi-Civita fields are the R-Urysohn ultrametric spaces
They extend Q and Q_p as fields while remaining metrically universal for separable ultrametrics
· “Algebraic structures on non-Archimedean Urysohn universal metric spaces”
Lindelöf strong gyrogroups are range-metrizable
Quotients keep Dugundji compacta and the weight-separability link under product closure
· “Characterizations of quotient spaces for Lindel\"of strongly topological gyrogroups”
Ray-space homeomorphism reduces to a checkable map between trees
It answers an open question and gives one game that also characterizes end, edge-end, and ultrametrizable spaces.
· “Ray and end spaces: characterizations and classification up to homeomorphism”
Crossing lines carry hidden dimension classical fractals miss
A new pointwise invariant detects directional coexistence at crossings, returning 2 where box dimension sees only 1
· “Point-dimension theory (part II): The point-cross dimension”
Banach lattices get Polish coding
Two faithful encodings of separable Banach lattices as Polish spaces, with the Fremlin tensor product proved Σ⁰₂-measurable and Gδ-graphed.
Diamond builds a Talagrand space that fails 2DCP
It keeps the Grothendieck and no-βω properties yet has no homeomorphic disjoint non-metrisable pieces
Rescaled norm upgrades bicomplex numbers to a Banach algebra
A factor-of-root-two rescaling eliminates the gap between modified and genuine Banach algebra structure on bicomplex spaces
· “Inner Products and Banach Algebra structures on Bicomplex Numbers and Their Associated Spaces”
Well-filteredness fails in lattices
Three counterexamples using Johnstone's dcpo resolve open problems on when well-filteredness is preserved under topology change and function
Lindelöf scattered W-spaces must be σ-compact
A topological game condition replaces external embedding, answering Tkachuk's question and tightening the link between covering and compact
Two forbidden retracts decide when a domain fails coherence
Bounded algebraic domains are non-coherent exactly when one of two model domains is a Scott-continuous retract.
Coherence and weak Hausdorffness lift together to Smyth powerspaces
The two properties are bonded: neither alone suffices, but together they transfer exactly, answering Xu's question on strong well-filtered.
Analytic height functions compactify to any mix of rays and circles
One non-finite vertex realises every prescribed combination of outgoing, incoming and circular edges
· “Compactifying real analytic functions and resulting Reeb spaces”
When probabilistic powerdomains stay RB-domains
No lower forks (and a root for probabilities) decide the property for every dcpo
· “A note on probabilistic powerdomains, RB-domains, and bc-domains”
Conjugate points can't stop transitive geodesic flows
The first examples of geodesic flows that stay transitive under all small perturbations despite having conjugate points.
· “Robust transitivity of geodesic flows from metrics with conjugate points”
Just two function spaces classify separable compact lines under BA
Dimension 0 or 1 fixes whether C_p(K) is I_B or I_B⊕I; another line has no square homeomorphism.
Cone domains are RB-domains only when their cone is simplicial
D_C is an RB-domain iff C is simplicial, giving FS-domains that are not RB-domains for any non-simplicial proper cone.
Poset powerdomain is RB-domain iff Hasse graph is tree with least element
Classification shows probabilistic powerdomain fails to preserve RB-domains, with diamond as counterexample.
· “Characterizing finite posets whose probabilistic powerdomain are RB-domains”
Homeomorphism groups of all pseudo-solenoids have non-metrizable flows
Even the pseudo-circle's homeomorphism group has a universal minimal flow that no metric can describe.
· “Universal minimal flows of the homeomorphism groups of pseudo-solenoids are non-metrizable”
Profiniteness equals residual finiteness for compact residuated lattices
The same equivalence holds for closed subdirect products of finite discrete structures under filter-induced linear topologies.
Weak ball of nonseparable Hilbert space homeomorphic to its positive part
The same ball is also homeomorphic to its product with the Hilbert cube, and all related B(κ,a,b) spaces coincide up to homeomorphism
· “Homeomorphism between close relatives of Hilbertian balls”
Countable Stone homeomorphism groups have just three coarse shapes
Each group is a point, a Hamming cube, or a tree-leaf set, decided by two ordinal invariants.
· “Coarse geometry of homeomorphism groups: Classifying countable Stone spaces”
Δ-spaces and Q-spaces match measurable cardinal strength
Equiconsistency settles consistency questions and bounds size of Lindelöf Q-spaces with weight at most the continuum.
Disk domain separates FS-domains from RB-domains
A 30-year open question in domain theory is settled by one concrete geometric counter-example
Halo operator equals ω-accumulation points on every space
The resulting operator satisfies axiom 4 without separation axioms and yields completeness of K4 for all infinite spaces.
646099441937791106493755218560442089979 labeled posets on 19 points
The 39-digit total extends the OEIS sequence A001035 and supplies the labeled topology count for 19 points.
· “The number of labeled partial orders and topologies on 19 points”
