Buried points of plane continua
Bibliographic record
Abstract
Abstract Sets on the boundary of a complementary component of a continuum in the plane have been of interest since the early 1920s. Curry and Mayer defined the buried points of a plane continuum to be the points in the continuum which were not on the boundary of any complementary component. Motivated by their investigations of Julia sets, they asked what happens if the set of buried points of a plane continuum is totally disconnected and nonempty. Curry, Mayer, and Tymchatyn showed that in that case the continuum is Suslinian, i.e., it does not contain an uncountable collection of nondegenerate pairwise disjoint subcontinua. In an answer to a question of Curry et al., van Mill and Tuncali constructed a plane continuum whose buried point set was totally disconnected, nonempty, and one-dimensional at each point of a countably infinite set. In this paper, we show that the van Mill–Tuncali example was the best possible in the sense that whenever the buried set is totally disconnected, it is one-dimensional at each of at most countably many points. As a corollary, we find that the buried set cannot be almost zero-dimensional unless it is zero-dimensional. We also construct locally connected van Mill–Tuncali type examples.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".