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.
Normal spaces and spaces, with the source disagreement over whether normality includes stated explicitly
Definition
Let be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).
- is normal when any two disjoint closed sets can be separated by disjoint open sets: for all closed with there are with
- is when it is normal and ( (Kolmogorov) and (Frechet) spaces).
Either of , may be empty, and those cases are met by or together with ; so the condition hides no nonemptiness hypothesis. As with regularity, "disjoint open sets" may equivalently be read as "disjoint open neighbourhoods of the two sets" (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).
Normality is the special case of complete normality at a disjoint closed pair. Disjoint closed sets are separated in the sense of Separated sets: , since the closure of a closed set is itself; so a space in which every separated pair can be put into disjoint open sets is in particular normal. That stronger condition is defined later on this page, and the implication is proved there.
The convention fork, and this library's side of it. Exactly as for regularity, textbooks disagree about whether normal carries a hypothesis. Munkres builds it in; Kelley, Willard and Engelking do not. This library takes the second side: normal names the separation condition alone, names normal plus , and the hypothesis is written out wherever it is used. The reason is again that the two halves are independent, and here the point is sharp: normality without implies nothing at all in the hierarchy. The indiscrete topology on a two-point set (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) is normal, its only closed sets being and the whole space, and it is not even ; Sierpinski space is normal, and not regular. Both are recorded on this page, the first as a false statement and both on the companion page.
Remarks
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Normality does not imply regularity, and the failure is witnessed by Sierpinski space on the companion page, which is normal and not regular. Whether regularity implies normality is a question this page leaves open: any witness reachable from the material here would need cardinal arithmetic or the hereditary behaviour of regularity. This page's own prerequisites still supply neither: cardinal arithmetic and cofinality is now built, but below this one, and nothing here draws on it; the hereditary and productive behaviour of the separation axioms is developed later in the reading order. So nothing above asserts an answer and no false statement asserting one is planted here (Conventions on this page, and the one implication of the classical chain that is not available at this point in the reading order).
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Normality is the axiom that behaves worst, and the companion page shows one symptom: the deleted Tychonoff plank, a subspace of a product of two ordinal spaces each of which is , is Hausdorff and not normal. Whether normality is inherited by subspaces or preserved by products is a question this page does not answer, and nothing here asserts an answer; the plank is presented only as a Hausdorff space that fails normality.
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What the definition does not say. It says nothing about separating a point from a closed set, because a point need not be closed; that is the content of the hypothesis in , and the theorem two items below is where it is spent.
Depends on
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- $T_0$ (Kolmogorov) and $T_1$ (Frechet) spaces
- Separated sets: $\overline{A} \cap B = A \cap \overline{B} = \varnothing$
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open
Used by
- Under dependent choice a normal T₁ space is completely regular, so T₄ ⟹ T_31/2, and together with the implications already proved this is the whole classical chain Corollary
- Under dependent choice, a continuous real-valued map on a closed subspace of a normal space extends to the whole space, and a map into an open interval extends into that same open interval Corollary
- In the K-topology on ℝ the closed set K ∪ {0} carries a continuous two-valued function with no continuous extension Counterexample
- Refuted, assuming countable choice: every Hausdorff space built from ordinal spaces is normal. The deleted Tychonoff plank ((ω₁ + 1) × (ω + 1)) ∖ {(ω₁, ω)} is Hausdorff and not normal Counterexample
- Regular and normal do not imply T₁ under the library's conventions Counterexample
- Completely normal (T₅) and perfectly normal (T₆) spaces Definition
- A continuous function on [0,1] ⊆ ℝ extended to all of ℝ, both by Tietze and by hand Example
- A discrete space satisfies every axiom in the chain; an indiscrete space with two points is regular, completely regular, normal, completely normal and perfectly normal, and fails T₀ Example
- A Urysohn function for (-∞, 0] and [1, ∞) in ℝ, written down and checked against the definition Example
- Sierpinski space is normal and not completely regular, so the T₁ hypothesis in the Urysohn corollary is not decoration Example
- Sierpinski space is T₀ and normal but neither T₁ nor regular: normality without T₁ implies nothing Example
- The cocountable topology on ℝ is T₁, has unique sequential limits, and is neither Hausdorff nor regular nor normal Example
- The cofinite topology on an infinite set is T₁ but neither Hausdorff nor regular nor normal Example
- The particular-point topology is T₀, it is not T₁ and not regular once the set has at least two points, and it is not normal once the set has at least three Example
- The sets U₀, U₁, U_1/2, U_1/4, U_3/4 of the Urysohn construction computed for two disjoint closed subsets of ℝ Example
- Assuming countable choice, refuted: every regular space is normal False statement
- FALSE: Every continuous real-valued function on a subspace of a normal space extends continuously to the whole space False statement
- FALSE: Every normal space is completely regular False statement
- FALSE: every normal space is Hausdorff, so the T₁ hypothesis in T₄ is redundant False statement
- Refuted: every paracompact space is normal False statement
- A space is normal if and only if every closed A inside an open U admits an open V with A ⊆ V ⊆ overlineV ⊆ U Lemma
- Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space Lemma
- Every closed subspace of a normal space is normal Lemma
- Every regular Lindelöf space is normal Lemma
- Jones's bound: under choice, a closed discrete subspace of a normal space cannot have more subsets than a dense set has subsets Lemma
- Conventions on this page, and the one implication of the classical chain that is not available at this point in the reading order Remark
- A compact Hausdorff space is regular and normal, hence T₃ and T₄ Theorem
- A normal T₁ space is regular, hence T₃, hence Urysohn, Hausdorff, T₁ and T₀ Theorem
- A space is completely normal if and only if every subspace is normal Theorem
- Assuming countable choice, every perfectly normal space is completely normal: separated sets in a normal space whose open sets are all F_σ can be separated by disjoint open sets Theorem
- Every completely normal space is normal, and every perfectly normal space is normal Theorem
- Every paracompact Hausdorff space is normal Theorem
- In a metric space any two separated sets have disjoint open neighbourhoods, so every metrizable space is completely normal Theorem
- The implications proved on this page: perfectly normal gives completely normal under countable choice, and completely normal gives normal; normal with T₁ gives T₃; completely regular gives regular; regular with T₁ gives Urysohn, hence Hausdorff, hence T₁, hence T₀; and metrizable gives every one of them Theorem
- Tietze's extension theorem, under dependent choice: a continuous map from a closed subspace of a normal space into [a,b] extends continuously to the whole space, and this property characterises normality Theorem
- Under dependent choice a space is perfectly normal if and only if it is normal and every closed set is a zero set Theorem
- Urysohn's lemma, under the axiom of dependent choice: in a normal space two disjoint closed sets are separated by a continuous function into [0,1], and conversely such a space is normal Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 45 results over 19 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
- Normal space (Wikipedia) (standard reference, not scraped)
- Separation axiom (Wikipedia) (standard reference, not scraped)
- J. Munkres, Topology, 2nd ed., §32 (standard reference, not scraped)
- R. Gardner, Introduction to Topology, notes on Munkres Section 31: The Separation Axioms (East Tennessee State University) (standard reference, not scraped)