WHEN THE HEWITT REALCOMPACTIFICATION AND THE P-COREFLECTION COMMUTE
Bibliographic record
Abstract
If X is a Tychonoff space then its P-coreflection Xδ is a Tychonoff space that is a dense subspace of the realcompact space (υX)δ, where υX denotes the Hewitt realcompactification of X. We investigate under what conditions Xδ is C-embedded in (υX)δ, i.e. under what conditions υ(Xδ) = (υX)δ. An example shows that this can fail for the product of a compact space and a P-space. It is possible for a von Neumann regular ring A to be isomorphic to a C(Y) and lie between C(X) and C(Xδ) without being isomorphic to C(Xδ). This cannot occur if X is realcompact or more generally if υ(Xδ) = (υX)δ. Applications are given to the epimorphic hull of C(X).
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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 | 0.001 | 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".