On structural numbers of topological spaces
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Bibliographic record
Abstract
Abstract Zero-dimensional structural numbers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> </m:mrow> <m:mn>0</m:mn> <m:mrow> <m:mtext>ind </m:mtext> </m:mrow> </m:msubsup> <m:mtext> and </m:mtext> <m:msubsup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> </m:mrow> <m:mn>0</m:mn> <m:mrow> <m:mtext>dim </m:mtext> </m:mrow> </m:msubsup> </m:math> $\mathcal{Z}_0^{\text {ind }} \text { and } \mathcal{Z}_0^{\text {dim }}$ w.r.t. dimensions ind and dim were introduced by Georgiou, Hattori, Megaritis, and Sereti. Somewhat similarly, we define structural numbers Sn 𝒜 for different subclasses 𝒜 of the class of hereditarily normal T 1 -spaces. In particular, if ℳ dim denotes the class of metrizable spaces Z with dim Z = 0 ; we show that: for any metrizable space X with dim X = n ≥ 0, we have <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mrow> <m:msup> <m:mo form="prefix">Sn</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mo form="prefix">dim</m:mo> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:mrow> <m:mi>X</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>;</m:mo> </m:math> $1 \leq \operatorname{Sn}^{\mathcal{M}_{\operatorname{dim}}} X \leq n+1 ;$ for any countable-dimensional metrizable space Y , we have <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mrow> <m:msup> <m:mo form="prefix">Sn</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mo form="prefix">dim</m:mo> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:mrow> <m:mi>Y</m:mi> <m:mo>≤</m:mo> <m:msub> <m:mi mathvariant="normal">ℵ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>.</m:mo> </m:math> $1 \leq \operatorname{Sn}^{\mathcal{M}_{\operatorname{dim}}} Y \leq \aleph_0 .$
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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 | 0.000 | 0.001 |
| 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) | 0.001 | 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