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.
, , , regularity, , complete regularity, and Tychonoffness are productive
Statement
Each of , , Hausdorffness (), regularity, , complete regularity, and Tychonoffness is productive.
Facts & Assumptions
Given: A family of spaces having one of the listed separation properties.
Arbitrary products preserve , , Hausdorffness, regularity, and complete regularity as stated in the three preceding lemmas (Arbitrary products preserve , , and Hausdorffness, Arbitrary products of regular spaces are regular, Arbitrary products of completely regular spaces are completely regular).
is regular plus , and Tychonoff is completely regular plus (Regular spaces and spaces, with the source disagreement over whether regularity includes stated explicitly, Completely regular spaces and Tychonoff () spaces).
Proof
The assertions for , , , regularity, and complete regularity are [L1].
A product of spaces is regular and by [L1], hence is .
A product of Tychonoff spaces is completely regular and by [L1], hence is Tychonoff.
Therefore every property in the statement is productive.
Depends on
- Arbitrary products preserve $T_0$, $T_1$, and Hausdorffness
- Arbitrary products of regular spaces are regular
- Arbitrary products of completely regular spaces are completely regular
- Regular spaces and $T_3$ spaces, with the source disagreement over whether regularity includes $T_1$ stated explicitly
- Completely regular spaces and Tychonoff ($T_{3\frac{1}{2}}$) spaces
Used by
- A worked preservation table for subspaces and products of the standard, cofinite, and ordinal topologies Example
- Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space Lemma
- Preservation ledger for the separation axioms, with T₁ conventions kept explicit Remark
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 72 results over 15 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
- J. P. May, An Outline Summary of Basic Point Set Topology, §6 (standard reference, not scraped)
- Separation axiom (Wikipedia) (standard reference, not scraped)