b∗-I-Open Sets and Their Role in Weaker Forms of Paracompactness
Bibliographic record
Abstract
This paper introduces and investigates two new notions of paracompactness in ideal topological spaces: \(b^*\)-\(\mathcal{I}\)-paracompactness and \(b_1^*\)-\(\mathcal{I}\)-paracompactness. These concepts generalize classical paracompactness using \(b^*\)-\(\mathcal{I}\)-open sets and ideal-related refinements. We establish fundamental properties of these spaces, including their preservation under subspaces, finite unions, and continuous mappings. Furthermore, we provide characterizations of these spaces and compare them with existing variants such as \(\beta\)-paracompactness, \(\beta_1\)-paracompactness and \(\mathcal{I}\)-paracompactness, supported by illustrative examples. Our results extend the theory of ideal topological spaces and offer a framework for future studies on generalized covering properties.
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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 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".