Hyperspace of finite unions of convergent sequences
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Bibliographic record
Abstract
The symbol S( X ) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X . This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S( X ). Then we consider some cardinal invariants on S( X ), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S ( X ) with the corresponding cardinal function of X . Moreover, we consider rank k -diagonal on S ( X ), and give a space X with a rank 2 -diagonal such that S ( X ) does not G δ -diagonal. Further, we study the relations of some generalized metric properties of X and its hyperspace S ( X ). Finally, we pose some questions about the hyperspace S ( X ).
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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