Bibliographic record
Abstract
A continuous map f : X → Y f:X\to Y of topological spaces X , Y X, Y is said to be almost 1 1 -to- 1 1 if the set of the points x ∈ X x\in X such that f − 1 ( f ( x ) ) = { x } f^{-1}(f(x))=\{x\} is dense in X X ; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and σ \sigma -compact spaces (e.g., n n -manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if f f is a minimal self-mapping of a 2-manifold M M , then point preimages under f f are tree-like continua and either M M is a union of 2-tori, or M M is a union of Klein bottles permuted by f f .
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.005 | 0.010 |
| Open science | 0.001 | 0.007 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.014 | 0.004 |
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".