How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Regular and normal do not imply under the library's conventions
Statement refuted
Every regular space, and every normal space, is .
Facts & Assumptions
Given: A two-point set with its indiscrete topology.
The indiscrete topology has only and as open sets, hence only and as closed sets (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).
Regularity separates a point from a closed set that misses it; normality separates disjoint closed sets; separates each distinct point from the other by an open set (Regular spaces and spaces, with the source disagreement over whether regularity includes stated explicitly, Normal spaces and spaces, with the source disagreement over whether normality includes stated explicitly, (Kolmogorov) and (Frechet) spaces).
Counterexample
In this topology the only closed set missing a point is , and it is separated from that point by and ; thus is regular.
Every disjoint pair of closed sets has an empty member, so the same two open sets show that is normal.
No open set contains one point while missing the other, so is not . This refutes both implications.
Depends on
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Regular spaces and $T_3$ spaces, with the source disagreement over whether regularity includes $T_1$ stated explicitly
- Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly
- $T_0$ (Kolmogorov) and $T_1$ (Frechet) spaces
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 48 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- S. Willard, General Topology, separation axioms (standard reference, not scraped)
- Sierpinski space (Wikipedia) (standard reference, not scraped)
- Separation axiom (Wikipedia) (standard reference, not scraped)