Linear and uniformly continuous surjections between <i>C<sub>p</sub> </i>-spaces over metrizable spaces
Why this work is in the frame
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Bibliographic record
Abstract
Abstract For any Tychonoff space X let D ( X ) be either the set C ( X ) of all continuous functions on X or the set C ∗ ( X ) of all bounded continuous functions on X . When D ( X ) is endowed with the pointwise convergence topology, we write D p ( X ). Let T : D p ( X ) → D p ( Y ) be a continuous linear surjection, where X is a metrizable space and Y is perfectly normal. We show that if X has some dimensional-like property 𝒫 , then so does Y . For example, 𝒫 could be one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. This result remains true if T is a uniformly continuous and inversely bounded surjection. Also, we consider other properties 𝒫 : of being a scattered space, or a strongly σ -scattered space, or a Δ1-space. Our results strengthen and extend several results from the various recently published papers.
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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 | 0.001 |
| 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 | 0.000 |
| 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