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Record W2313510692 · doi:10.4153/cjm-2012-022-4

Distance Sets of Urysohn Metric Spaces

2012· article· en· W2313510692 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueCanadian Journal of Mathematics · 2012
Typearticle
Languageen
FieldMathematics
TopicAdvanced Topology and Set Theory
Canadian institutionsUniversity of Calgary
Fundersnot available
KeywordsMathematicsInjective metric spaceMetric spaceConvex metric spaceIntrinsic metricMetric differentialIsometry (Riemannian geometry)Metric (unit)Separable spaceUrysohn and completely Hausdorff spacesSubspace topologySpace (punctuation)CombinatoricsCardinality (data modeling)Discrete mathematicsPure mathematicsMathematical analysisHausdorff dimensionComputer science

Abstract

fetched live from OpenAlex

Abstract. A metric space M = ( M ; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M). A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist( U ).) The main results are: (1) A characterization of the sets dist( U ) for Urysohn metric spaces U . (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R , then they amalgamate disjointly to a metric space with distances in R . (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.002
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.032
Threshold uncertainty score0.444

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.002
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.039
GPT teacher head0.306
Teacher spread0.267 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it