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Normal space
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Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T3½(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
 Hausdorff)
T6 (perfectly normal
 Hausdorff)

In topology and related branches of mathematics, a normal space is a topological space in which any two disjoint closed sets have disjoint open neighborhoods. Such spaces need not be Hausdorff in general. A normal Hausdorff space is called a T4 space. Strengthenings of these concepts are detailed in the article below and include completely normal spaces and perfectly normal spaces, and their Hausdorff variants: T5 spaces and T6 spaces. All these conditions are examples of separation axioms.

Definitions

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A topological space X is a normal space if, given any disjoint closed sets E and F, there are neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.

The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods U and V, here represented by larger, but still disjoint, open disks.

A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.

A completely normal space, or hereditarily normal space, is a topological space X such that every subspace of X is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods. Also, X is completely normal if and only if every open subset of X is normal with the subspace topology.

A T5 space, or completely T4 space, is a completely normal T1 space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T4 space.

A perfectly normal space is a topological space in which every two disjoint closed sets and can be precisely separated by a function, in the sense that there is a continuous function from to the interval such that and .[1] This is a stronger separation property than normality, as by Urysohn's lemma disjoint closed sets in a normal space can be separated by a function, in the sense of and , but not precisely separated in general. It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X is perfectly normal if and only if every closed set is the zero set of a continuous function. The equivalence between these three characterizations is called Vedenissoff's theorem.[2][3] Every perfectly normal space is completely normal, because perfect normality is a hereditary property.[4][5]

A T6 space, or perfectly T4 space, is a perfectly normal Hausdorff space.

Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever the meaning of T4 may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.

Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".

Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.

A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Nemytskii plane.

Examples of normal spaces

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Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:

Also, all fully normal spaces are normal (even if not regular). Sierpiński space is an example of a normal space that is not regular.

Examples of non-normal spaces

[edit]

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.

A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the product of uncountably many non-compact metric spaces is never normal.

Properties

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Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.[6]

The main significance of normal spaces lies in the fact that they admit "enough" continuous real-valued functions, as expressed by the following theorems valid for any normal space X.

Urysohn's lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from X to the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to be entirely within the unit interval [0,1]. In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also separated by a function.

More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A to R, then there exists a continuous function F: X → R that extends f in the sense that F(x) = f(x) for all x in A.

The map has the lifting property with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.[7]

If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U. This shows the relationship of normal spaces to paracompactness.

In fact, any space that satisfies any one of these three conditions must be normal.

A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example of this phenomenon is the Sorgenfrey plane. In fact, since there exist spaces which are Dowker, a product of a normal space and [0, 1] need not to be normal. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (Tychonoff's theorem) and the T2 axiom are preserved under arbitrary products.[8]

Relationships to other separation axioms

[edit]

If a normal space is R0, then it is in fact completely regular. Thus, anything from "normal R0" to "normal completely regular" is the same as what we usually call normal regular. Taking Kolmogorov quotients, we see that all normal T1 spaces are Tychonoff. These are what we usually call normal Hausdorff spaces.

A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.

Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpiński space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.

See also

[edit]

Citations

[edit]
  1. ^ Willard, Exercise 15C
  2. ^ Engelking, Theorem 1.5.19. This is stated under the assumption of a T1 space, but the proof does not make use of that assumption.
  3. ^ "Why are these two definitions of a perfectly normal space equivalent?".
  4. ^ Engelking, Theorem 2.1.6, p. 68
  5. ^ Munkres 2000, p. 213
  6. ^ Willard 1970, pp. 100–101.
  7. ^ "separation axioms in nLab". ncatlab.org. Retrieved 2021-10-12.
  8. ^ Willard 1970, Section 17.

References

[edit]
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Normal space

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In , a normal space is a XX in which, for any two disjoint closed subsets AA and BB of XX, there exist disjoint open subsets UU and VV of XX such that A⊆UA \subseteq U and B⊆VB \subseteq V. This separation property strengthens the axioms for and is often denoted as the T4T_4 when combined with the T1T_1 condition (where singletons are closed sets, equivalent to the Hausdorff property). The definition typically assumes T1T_1, ensuring that points are closed and that the space distinguishes distinct points with disjoint open neighborhoods. Normal spaces form a key part of the hierarchy of separation axioms in , where every normal space is regular (disjoint closed sets and points can be separated by open sets) and every is Hausdorff, with these inclusions being proper. A fundamental consequence is , which states that in a normal space, for any disjoint closed sets AA and BB, there exists a f:X→[0,1]f: X \to [0,1] such that f(A)={0}f(A) = \{0\} and f(B)={1}f(B) = \{1\}, enabling the embedding of such spaces into metric-like structures under additional conditions. The further extends this, allowing continuous real-valued functions defined on closed subspaces to be extended to the entire space. Notable examples of normal spaces include all metric spaces equipped with their standard topology, as the distance function allows separation of closed sets by open balls. Compact Hausdorff spaces are also normal, as are well-ordered sets under the and regular second-countable spaces. However, normality is not preserved under arbitrary products or subspaces; for instance, the uncountable product of intervals (0,1)J(0,1)^J for uncountable JJ is regular but not normal, and the Sorgenfrey plane (product of Sorgenfrey lines) fails normality despite each factor being normal. These properties make normal spaces essential in metrization theorems, such as the Urysohn metrization theorem, which characterizes second-countable normal T1T_1 spaces as metrizable.

Core Definitions

Formal Definition

In topology, a closed set in a topological space is the complement of an open set. A topological space XX is called normal if, given any two disjoint closed subsets AA and BB of XX, there exist disjoint open subsets UU and VV of XX such that A⊆UA \subseteq U and B⊆VB \subseteq V. A normal space that is also T1T_1—meaning that it satisfies the where singleton sets are closed, allowing points to be separated from closed sets by open neighborhoods—is denoted a T4T_4 space.

Equivalent Formulations

In T1T_1 spaces, every normal is regular, with normality extending the separation property from points and disjoint closed sets to arbitrary pairs of disjoint closed sets. An alternative characterization states that a topological space is normal if and only if, for every closed set EE and every open set UU containing EE, there exists an open set VV such that E⊆V⊆V‾⊆UE \subseteq V \subseteq \overline{V} \subseteq U. Normality can also be defined in terms of function extension: a space is normal if and only if every continuous real-valued function defined on a closed subset extends to a continuous function on the entire space. This is known as Tietze's extension theorem. Urysohn's lemma provides another equivalent formulation: a space is normal if and only if, for any two disjoint closed sets EE and FF, there exists a continuous function f:X→[0,1]f: X \to [0,1] such that f(E)={0}f(E) = \{0\} and f(F)={1}f(F) = \{1\}. The concept of normal spaces was introduced by Felix Hausdorff in 1914, playing a pivotal role in the early development of general topology by formalizing higher separation axioms beyond Hausdorff spaces.

Examples

Normal Spaces

Metric spaces provide a fundamental class of normal topological spaces. In a metric space (X,d)(X, d), any two disjoint closed sets AA and BB can be separated by open sets constructed using open balls of radius equal to half the infimum of the distances between points in AA and BB, ensuring the openness and disjointness required by the normality axiom. This property holds because the metric induces a topology where such balls form a basis, allowing precise control over neighborhoods around closed sets. Compact s also satisfy normality. Every compact Hausdorff space is normal, as the compactness ensures that closed subsets are compact and can be separated using the Hausdorff property combined with finite subcovers to construct disjoint open neighborhoods. further supports this by guaranteeing that products of compact Hausdorff spaces remain compact and Hausdorff, preserving normality. Euclidean spaces Rn\mathbb{R}^n exemplify normal spaces through their metric structure, inheriting the separation properties of metric spaces directly from the Euclidean metric. Similarly, topological manifolds, being locally Euclidean and Hausdorff with a second-countable , are normal due to their metrizable local charts that extend globally via the manifold's structure. Finite products of normal spaces are normal in the . For instance, the product of two normal spaces inherits the because projections allow lifting separations from each factor to disjoint open sets in the product. Infinite products under the Tychonoff can be normal if the factors are compact, as the resulting space is compact Hausdorff by . A example is the unit interval [0,1][0,1] with the standard , which is compact and Hausdorff, hence normal. This space satisfies the T4 , combining normality with the T1 property inherent to Hausdorff spaces.

Non-Normal Spaces

The Niemytzki plane, also known as the Moore plane, is defined as the upper half-plane R×[0,∞)\mathbb{R} \times [0, \infty) equipped with a topology where the basis consists of all open disks in the upper half-plane for points with positive yy-coordinate, and for points on the xx-axis (y=0y=0), the basis elements are open disks tangent to the xx-axis at that point and lying entirely in the upper half-plane. This space is Hausdorff and completely regular but fails to be normal. Specifically, let A=Q×{0}A = \mathbb{Q} \times \{0\} be the set of points on the xx-axis with rational xx-coordinates, and B=(R∖Q)×{0}B = (\mathbb{R} \setminus \mathbb{Q}) \times \{0\} the points with irrational xx-coordinates; both AA and BB are closed and disjoint in the Niemytzki plane, yet there do not exist disjoint open sets containing them, as any open neighborhood of a point in AA intersects every open neighborhood of nearby points in BB due to the tangent disk basis. The Sorgenfrey line is the real line R\mathbb{R} with the , generated by basis elements [a,b)[a, b) for a<ba < b. This space is normal, hereditarily Lindelöf, and paracompact, but its product with itself, the Sorgenfrey plane Rl×Rl\mathbb{R}_l \times \mathbb{R}_l, is not normal. The failure arises from the sets P={(p,−p)∣p∈R∖Q}P = \{(p, -p) \mid p \in \mathbb{R} \setminus \mathbb{Q}\} (the anti-diagonal over irrationals) and Q = \{(q, -q) \mid q \in \mathbb{Q}\}&#36; (over rationals), which are both closed and disjoint; however, they cannot be separated by disjoint open sets because any basic open neighborhood in the product topology around points in PandandQ$ will overlap due to the half-open intervals aligning along the anti-diagonal. The Tychonoff plank is the product space ([0,ω1]×[0,ω])([0, \omega_1] \times [0, \omega]) equipped with the , where ω1\omega_1 is the first uncountable ordinal and ω\omega is the first infinite ordinal. The deleted Tychonoff plank, obtained by removing the point (ω1,ω)(\omega_1, \omega), is completely regular but not normal. In this space, the sets C={ω1}×[0,ω)C = \{\omega_1\} \times [0, \omega) and D=[0,ω1)×{ω}D = [0, \omega_1) \times \{\omega\} are closed and disjoint, but no disjoint open sets separate them, as any open neighborhood of CC must include points arbitrarily close to (ω1,ω)(\omega_1, \omega) from below in the second coordinate, which inevitably intersects neighborhoods of DD near the deleted point. These examples illustrate pathologies that do not occur in metric spaces, which are always normal.

Key Properties

Separation Properties

A fundamental separation property of normal spaces is encapsulated in , which states that if XX is a normal space and A,B⊆XA, B \subseteq X are disjoint closed sets, then there exists a f:X→[0,1]f: X \to [0,1] such that f(A)={0}f(A) = \{0\} and f(B)={1}f(B) = \{1\}. Intuitively, this lemma provides a continuous "separator" that distinguishes the two closed sets by mapping one to the zero level and the other to the unit level, with intermediate values ensuring continuity across the space; it relies on the T1 condition to ensure singletons are closed, making the property hold specifically for T4 spaces (normal and T1). The extends this separability to real-valued functions: in a normal space XX, if A⊆XA \subseteq X is closed and g:A→Rg: A \to \mathbb{R} is continuous, then there exists a continuous extension G:X→RG: X \to \mathbb{R} such that G∣A=gG|_A = g. This theorem underscores the flexibility of normal spaces in extending local continuous data globally while preserving the function's boundedness if the original is bounded. In paracompact T4 spaces, every open cover admits a locally finite open refinement to which is subordinate, consisting of continuous functions {ϕi}\{\phi_i\} such that each ϕi≥0\phi_i \geq 0, ∑ϕi=1\sum \phi_i = 1, and supp⁡(ϕi)\operatorname{supp}(\phi_i) is contained in the corresponding refinement set. Compact Hausdorff spaces, being normal, are paracompact, meaning every open cover has a locally finite open refinement. Moreover, a normal space that is also paracompact satisfies metrizability under additional conditions, such as possessing a countable basis.

Embedding and Extension Theorems

Normal spaces, being completely regular, admit a canonical embedding into a product of closed intervals. Specifically, every normal space XX is homeomorphic to a subspace of the Tychonoff cube [0,1]I[0,1]^I for some II, where the is constructed using a family of continuous functions from XX to [0,1][0,1] that separate points from closed sets. This theorem, originally established by Tychonoff, highlights the functional richness of normal spaces and facilitates their study within the broader class of completely regular spaces. A key extension theorem for normal spaces arises from their complete regularity, enabling the construction of the Stone-Čech compactification βX\beta X. For any normal space XX, βX\beta X is a compact into which XX embeds densely as a subspace, and every continuous bounded real-valued function on XX extends uniquely to a on βX\beta X. This compactification, developed independently by Stone and Čech, preserves the topological structure of XX while extending it to a compact setting, making it invaluable for analyzing limits and extensions in normal spaces. Metrization theorems provide conditions under which normal spaces admit a compatible metric, thereby embedding them into metric spaces. The Urysohn metrization theorem states that every second-countable normal space is metrizable, as normality implies the required regularity and Hausdorff properties. This result, due to Urysohn, is particularly powerful for spaces with countable bases, such as manifolds or separable spaces, allowing the transfer of metric tools like completeness and . For compact normal spaces, metrizability follows under second countability, strengthening the theorem's applicability in bounded settings where compactness ensures normality. The Bing metrization theorem extends these ideas to a broader class of normal spaces without assuming second countability. It asserts that a normal space is metrizable it has a basis, meaning a basis that is a countable union of discrete families of open sets. Named after Bing, this theorem characterizes metrizability through base conditions that align with normality's separation capabilities, enabling metrization for certain locally compact or paracompact normal spaces beyond the second-countable case.

Relations to Separation Axioms

Comparisons with T₀ to T₃ Axioms

In , the separation axioms T₀ through T₄ form a of increasingly stringent conditions on how well points and sets can be distinguished using open sets in a . A normal space, often denoted as satisfying T₄ when combined with T₁, implies all weaker axioms in this chain, providing a structured progression from basic point separation to the separation of disjoint closed sets. The weakest axiom, T₀ (also known as Kolmogorov quotient), requires that for any two distinct points xx and yy in the , there exists an open set containing one but not the other. This ensures a minimal level of distinguishability between points. Every normal satisfies T₀, as the stronger separation properties guarantee such open sets exist. Next, T₁ (Fréchet ) strengthens T₀ by requiring that every singleton set {x}\{x\} is closed, which is equivalent to the condition that for distinct points xx and yy, there are open sets containing xx but not yy, and vice versa. Normality requires T₁ for the T₄ designation, as the definition of T₄ explicitly includes T₁ alongside the normality condition for separating disjoint closed sets. T₂ (Hausdorff space) further refines separation by demanding that any two distinct points have disjoint open neighborhoods. A normal space implies T₂ provided it also satisfies T₁, since the ability to separate points from closed sets and closed sets from each other cascades to point-point separation. T₃ (regular Hausdorff space) combines regularity—where a point and a disjoint have disjoint open neighborhoods—with T₁ (or sometimes T₀ in alternative conventions). A T₄ space is precisely a T₃ space augmented by the condition that any two disjoint can be separated by disjoint open neighborhoods. The implications form a strict chain: T₄ ⇒ T₃ ⇒ T₂ ⇒ T₁ ⇒ T₀, where each step adds a layer of separation power, though the reverse implications do not hold. For instance, there exist spaces that are T₃ but not T₄, satisfying point-closed set separation yet failing to separate certain pairs of disjoint closed sets, and similarly for weaker levels relative to T₄.

Stronger Notions of Normality

A completely normal space is defined as a where every pair of separated sets possesses disjoint open neighborhoods. Two sets AA and BB are separated if A∩B‾=∅A \cap \overline{B} = \emptyset and B∩A‾=∅B \cap \overline{A} = \emptyset. This condition strengthens the normality axiom by ensuring separation not just for disjoint closed sets but for sets that are mutually disjoint from each other's closures. Completely normal spaces imply normality, as disjoint closed sets are a special case of , but the converse does not hold. For instance, all metric spaces are completely normal, since subspaces of metric spaces are metrizable and thus normal. In contrast, the Tychonoff plank—defined as ([0,ω1]×[0,ω])([0, \omega_1] \times [0, \omega]) with the product —is normal but not completely normal, as the deleted Tychonoff plank (removing the point (ω1,ω)(\omega_1, \omega)) is a non-normal subspace. Equivalently, the subspace consisting of the "residual boundaries" {ω1}×[0,ω)∪([0,ω1)×{ω}\{ \omega_1 \} \times [0, \omega) \cup ([0, \omega_1) \times \{ \omega \} fails to be normal in the subspace topology, since the components {ω1}×[0,ω)\{ \omega_1 \} \times [0, \omega) and ([0,ω1)×{ω}([0, \omega_1) \times \{ \omega \} are disjoint closed sets that cannot be separated by disjoint open sets. Hereditarily normal spaces provide an equivalent formulation to complete normality: a space is hereditarily normal if every subspace, endowed with the , is normal. This equivalence arises because the separation property for ensures that induced topologies on arbitrary subspaces preserve normality. Metric spaces exemplify hereditarily normal spaces, inheriting this property from their metrizability. The Tychonoff plank serves as a , being normal yet containing a non-normal subspace as noted above. Monotonically normal spaces extend normality in spaces with order structure, requiring a monotone separation operator for disjoint closed sets. Specifically, for disjoint closed sets AA and BB, there exists a function UU assigning to each such pair an U(A,B)U(A, B) containing AA and disjoint from BB, such that if A′⊆AA' \subseteq A and B′⊆BB' \subseteq B are disjoint closed sets, then U(A′,B′)⊆U(A,B)U(A', B') \subseteq U(A, B). This monotonicity ensures consistent refinement of neighborhoods. Every generalized ordered space (GO-space), including linearly ordered topological spaces, is monotonically normal, highlighting its relevance to ordered topologies. Collectionwise normal spaces strengthen normality by handling families of closed sets simultaneously. A T1T_1 space is collectionwise normal if, for every discrete collection {Fi}i∈I\{F_i\}_{i \in I} of closed sets, there exists a discrete collection {Ui}i∈I\{U_i\}_{i \in I} of open sets such that Fi⊆UiF_i \subseteq U_i for each ii and the UiU_i are pairwise disjoint. This property exceeds mere pairwise separation and is crucial in dimension theory, where it facilitates inductive constructions for embedding spaces into Euclidean spaces. Examples include all compact metric spaces, while certain Moore planes provide counterexamples to weaker forms. Completely normal spaces that are also Lindelöf are paracompact, as the Lindelöf property combined with normality (which complete normality implies under T1T_1) yields a locally finite open refinement for every open cover via standard theorems on regular Lindelöf spaces.

References

  1. https://mathworld.wolfram.com/NormalSpace.html
  2. https://ncatlab.org/nlab/show/normal%2Bspace
  3. https://faculty.etsu.edu/gardnerr/5357/notes/Munkres-31.pdf
  4. https://faculty.etsu.edu/gardnerr/5357/notes/Munkres-32.pdf
  5. https://web.math.utk.edu/~freire/teaching/m467f19/UrysohnMetrization.pdf
  6. https://mathworld.wolfram.com/SeparationAxioms.html
  7. https://mathworld.wolfram.com/T1-SeparationAxiom.html
  8. https://mathworld.wolfram.com/RegularSpace.html
  9. https://www.math.wustl.edu/~freiwald/topology/Ch7.pdf
  10. https://archive.org/details/grundzgedermen00hausuoft
  11. https://www.math.utoronto.ca/ivan/mat327/docs/notes/12-metric.pdf
  12. https://www.math.auckland.ac.nz/~gauld/750-05/section4.pdf
  13. https://people.math.sc.edu/nyikos/Manifolds.pdf
  14. https://researchrepository.wvu.edu/cgi/viewcontent.cgi?article=1002&context=math-grad-exams
  15. https://www.jstor.org/stable/24340013
  16. https://mizar.uwb.edu.pl/fm/2013-21/pdf21-2/topgen_6.pdf
  17. http://ericmoorhouse.org/handouts/tychonoff_plank.pdf
  18. https://eudml.org/doc/159111
  19. https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-1/issue-3/An-extension-of-Tietzes-theorem/pjm/1103052106.full
  20. https://webspace.science.uu.nl/~crain101/topologie11/chapter5.pdf
  21. https://mathworld.wolfram.com/CompletelyRegularSpace.html
  22. https://www.math.toronto.edu/ivan/mat327/docs/notes/09-separation.pdf
  23. http://webhome.auburn.edu/~gruengf/7500c.pdf
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  25. https://www.ejpam.com/index.php/ejpam/article/view/6597/2455
  26. https://web.mat.bham.ac.uk/C.Good/research/pdfs/ency-lind.pdf
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