Bibliographic record
Abstract
Let \( S \) be any proper subset of a topological space \( X \). We can introduce a finer topology \( \rho^{S} \) on \( X \) by designating all singletons in \( X \setminus S \) as open subsets. Each point in \( S \) retains the same open neighborhoods as in the original topology. We define a function \( \jmath \) as a neighborhood assignment or operator if it maps elements from \( X \) to the topology of \( X \), associates pairs of ordered disjoint closed subsets to the topology of \( X \), or links pairs \( (x, U) \), where \( U \) is an open neighborhood of \( x \), to the space \( X \). The space \( X \) is termed monotonically normal if there exists an \( M- \) operator on \( X \) that satisfies specific criteria. Our findings reveal that if \( X \) possesses the property of being monotonically normal, then for any proper subset \( S \) of \( X \), the discrete extension space \( X^{S} \) is also monotonically normal. Furthermore, we demonstrate that for a given topological space \( X \) and any finite subset \( S \subset X \), the discrete extension \( X^{S} \) achieves monotonic normality if either \( S \subset F_{1} \) or all elements of \( S \) lie outside \( F_{1} \) for every ordered pair \( (F_{1}, F_{2}) \) of disjoint closed subsets. Our exploration also examines the interplay between this type of extension and the concept of \( D- \) spaces. Notably, we establish that if \( X \) is a topological space and \( S \) is a compact proper subset of \( X \) such that \( X^{S} \) is discretely complete, then \( X^{S} \) qualifies as a \( D- \) space provided that \( X \setminus S \) is locally finite.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".