Tri-Expandability and Product Spaces in Tri-Topological Spaces
Why this work is in the frame
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Bibliographic record
Abstract
In this paper, we introduce and investigate the notion of tri-expandability in tri-topological spaces as a natural generalization of expandability in classical topological spaces. We establish fundamental characterizations of tri-expandable spaces and explore their behavior under product operations. The main results include a comprehensive study of the relationships between various forms of tri-expandability and their connections to classical expandability properties. We prove that tri-expandability is preserved under certain product constructions and provide necessary and sufficient conditions for a tri-topological space to be tri-expandable. Our findings extend the classical theory of expandable spaces to the multi-topological setting and reveal new structural properties that are unique to the tri-topological framework. Several illustrative examples demonstrate the richness of the theory and highlight the differences from the classical case.
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Full frame distilled prediction
Teacher imitationNot 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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it