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 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 | 0.001 |
| 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 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".