Lξ-Families: Localized Topology with Applications in Edge Detection
Bibliographic record
Abstract
We introduce and explore Lξ-families, an innovative class of localized topological structures that extends classical concepts while preserving fundamental properties. These families constitute a bridge between traditional topological objects and finer-grained local-to-global characteristics. Our construction offers a natural generalization of regular open sets through a novel localization approach that maintains critical topological invariants across various transformations and operations. This paper establishes the foundational theory of Lξ-families, proving key characterization theorems and situating them within the broader topological landscape. Our findings reveal that these structures form a complete lattice under appropriate operations and possess significant hereditary characteristics. Additionally, we demonstrate stability properties under continuous mappings and homeomorphisms, highlighting their seamless integration with established topological frameworks. Through strategically selected counterexamples, we define the boundaries of these new concepts. The theoretical architecture developed in this work creates pathways for applications in digital topology and image processing, with particularly promising implications for edge detection and boundary analysis methods. The relationships we establish between Lξ family members and classical topological concepts provide unifying perspectives across seemingly disparate notions and introduce novel tools for topological classification challenges.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".