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Normal space

In topology, a normal space is a topological space X in which, for any two disjoint closed subsets A and B of X, there exist disjoint open subsets U and V of X such that A \subseteq U and B \subseteq V.^[1] This separation property strengthens the axioms for topological spaces and is often denoted as the T_4 separation axiom when combined with the T_1 condition (where singletons are closed sets, equivalent to the Hausdorff property).^[2] The definition typically assumes T_1, ensuring that points are closed and that the space distinguishes distinct points with disjoint open neighborhoods.^[3] Normal spaces form a key part of the hierarchy of separation axioms in general topology, where every normal space is regular (disjoint closed sets and points can be separated by open sets) and every regular space is Hausdorff, with these inclusions being proper.^[3] A fundamental consequence is Urysohn's lemma, which states that in a normal space, for any disjoint closed sets A and B, there exists a continuous function f: X \to [0,1] such that f(A) = \{0\} and f(B) = \{1\}, enabling the embedding of such spaces into metric-like structures under additional conditions.^[2] The Tietze extension theorem further extends this, allowing continuous real-valued functions defined on closed subspaces to be extended to the entire space.^[2] 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.^[2] Compact Hausdorff spaces are also normal, as are well-ordered sets under the order topology and regular second-countable spaces.^[4] However, normality is not preserved under arbitrary products or subspaces; for instance, the uncountable product of intervals (0,1)^J for uncountable J is regular but not normal, and the Sorgenfrey plane (product of Sorgenfrey lines) fails normality despite each factor being normal.^[4] These properties make normal spaces essential in metrization theorems, such as the Urysohn metrization theorem, which characterizes second-countable normal T_1 spaces as metrizable.^[5]

Core Definitions

Formal Definition

In topology, a closed set in a topological space is the complement of an open set. A topological space X is called normal if, given any two disjoint closed subsets A and B of X, there exist disjoint open subsets U and V of X such that A \subseteq U and B \subseteq V.^[1] A normal space that is also T_1—meaning that it satisfies the separation axiom where singleton sets are closed, allowing points to be separated from closed sets by open neighborhoods—is denoted a T_4 space.^[6]^[7]

Equivalent Formulations

In T_1 spaces, every normal space is regular, with normality extending the separation property from points and disjoint closed sets to arbitrary pairs of disjoint closed sets.^[8] An alternative characterization states that a topological space is normal if and only if, for every closed set E and every open set U containing E, there exists an open set V such that E \subseteq V \subseteq \overline{V} \subseteq U.^[9] 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.^[9] This is known as Tietze's extension theorem.^[9] Urysohn's lemma provides another equivalent formulation: a space is normal if and only if, for any two disjoint closed sets E and F, there exists a continuous function f: X \to [0,1] such that f(E) = \{0\} and f(F) = \{1\}.^[9] 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.^[10]

Examples

Normal Spaces

Metric spaces provide a fundamental class of normal topological spaces. In a metric space (X, d), any two disjoint closed sets A and B can be separated by open sets constructed using open balls of radius equal to half the infimum of the distances between points in A and B, ensuring the openness and disjointness required by the normality axiom.^[11] This property holds because the metric induces a topology where such balls form a basis, allowing precise control over neighborhoods around closed sets.^[4] Compact Hausdorff spaces 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.^[12] Tychonoff's theorem further supports this by guaranteeing that products of compact Hausdorff spaces remain compact and Hausdorff, preserving normality.^[4] Euclidean spaces \mathbb{R}^n exemplify normal spaces through their metric structure, inheriting the separation properties of metric spaces directly from the Euclidean metric.^[11] Similarly, topological manifolds, being locally Euclidean and Hausdorff with a second-countable topology, are normal due to their metrizable local charts that extend globally via the manifold's structure.^[13] Finite products of normal spaces are normal in the product topology. For instance, the product of two normal spaces inherits the separation axiom because projections allow lifting separations from each factor to disjoint open sets in the product.^[14] Infinite products under the Tychonoff topology can be normal if the factors are compact, as the resulting space is compact Hausdorff by Tychonoff's theorem.^[4] A concrete example is the unit interval [0,1] with the standard topology, which is compact and Hausdorff, hence normal.^[12] This space satisfies the T4 separation axiom, combining normality with the T1 property inherent to Hausdorff spaces.^[4]

Non-Normal Spaces

The Niemytzki plane, also known as the Moore plane, is defined as the upper half-plane \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 y-coordinate, and for points on the x-axis (y=0), the basis elements are open disks tangent to the x-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 = \mathbb{Q} \times \{0\} be the set of points on the x-axis with rational x-coordinates, and B = (\mathbb{R} \setminus \mathbb{Q}) \times \{0\} the points with irrational x-coordinates; both A and B 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 A intersects every open neighborhood of nearby points in B due to the tangent disk basis.^[15] The Sorgenfrey line is the real line \mathbb{R} with the lower limit topology, generated by basis elements [a, b) for a < b. This space is normal, hereditarily Lindelöf, and paracompact, but its product with itself, the Sorgenfrey plane \mathbb{R}_l \times \mathbb{R}_l, is not normal. The failure arises from the sets 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

PandQ$ will overlap due to the half-open intervals aligning along the anti-diagonal.^[16] The Tychonoff plank is the product space ([0, \omega_1] \times [0, \omega]) equipped with the order topology, where \omega_1 is the first uncountable ordinal and \omega is the first infinite ordinal. The deleted Tychonoff plank, obtained by removing the point (\omega_1, \omega), is completely regular but not normal. In this space, the sets C = \{\omega_1\} \times [0, \omega) and D = [0, \omega_1) \times \{\omega\} are closed and disjoint, but no disjoint open sets separate them, as any open neighborhood of C must include points arbitrarily close to (\omega_1, \omega) from below in the second coordinate, which inevitably intersects neighborhoods of D near the deleted point.^[17] 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 Urysohn's lemma, which states that if X is a normal space and A, B \subseteq X are disjoint closed sets, then there exists a continuous function f: X \to [0,1] such that f(A) = \{0\} and f(B) = \{1\}.^[18] 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).^[18] The Tietze extension theorem extends this separability to real-valued functions: in a normal space X, if A \subseteq X is closed and g: A \to \mathbb{R} is continuous, then there exists a continuous extension G: X \to \mathbb{R} such that G|_A = g.^[19] 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 a partition of unity is subordinate, consisting of continuous functions \{\phi_i\} such that each \phi_i \geq 0, \sum \phi_i = 1, and \operatorname{supp}(\phi_i) is contained in the corresponding refinement set.^[20] 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 X is homeomorphic to a subspace of the Tychonoff cube [0,1]^I for some index set I, where the embedding is constructed using a family of continuous functions from X to [0,1] that separate points from closed sets.^[21] This embedding 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 \beta X. For any normal space X, \beta X is a compact Hausdorff space into which X embeds densely as a subspace, and every continuous bounded real-valued function on X extends uniquely to a continuous function on \beta X. This compactification, developed independently by Stone and Čech, preserves the topological structure of X 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 uniform continuity. 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 if and only if it has a \sigma-discrete 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 topology, the separation axioms T₀ through T₄ form a hierarchy of increasingly stringent conditions on how well points and sets can be distinguished using open sets in a topological space. 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.^[3] The weakest axiom, T₀ (also known as Kolmogorov quotient), requires that for any two distinct points x and y in the space, there exists an open set containing one but not the other. This ensures a minimal level of distinguishability between points. Every normal space satisfies T₀, as the stronger separation properties guarantee such open sets exist.^[3]^[22] Next, T₁ (Fréchet space) strengthens T₀ by requiring that every singleton set \{x\} is closed, which is equivalent to the condition that for distinct points x and y, there are open sets containing x but not y, 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.^[3]^[22] 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.^[3]^[22] T₃ (regular Hausdorff space) combines regularity—where a point and a disjoint closed set 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 closed sets can be separated by disjoint open neighborhoods.^[3]^[22] 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.^[3]^[22]

Stronger Notions of Normality

A completely normal space is defined as a topological space where every pair of separated sets possesses disjoint open neighborhoods. Two sets A and B are separated if A \cap \overline{B} = \emptyset and 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.^[23] Completely normal spaces imply normality, as disjoint closed sets are a special case of separated sets, 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, \omega_1] \times [0, \omega]) with the product order topology—is normal but not completely normal, as the deleted Tychonoff plank (removing the point (\omega_1, \omega)) is a non-normal subspace. Equivalently, the subspace consisting of the "residual boundaries" \{ \omega_1 \} \times [0, \omega) \cup ([0, \omega_1) \times \{ \omega \} fails to be normal in the subspace topology, since the components \{ \omega_1 \} \times [0, \omega) and ([0, \omega_1) \times \{ \omega \} are disjoint closed sets that cannot be separated by disjoint open sets.^[17] Hereditarily normal spaces provide an equivalent formulation to complete normality: a space is hereditarily normal if every subspace, endowed with the subspace topology, is normal. This equivalence arises because the separation property for separated sets 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 counterexample, being normal yet containing a non-normal subspace as noted above.^[23] Monotonically normal spaces extend normality in spaces with order structure, requiring a monotone separation operator for disjoint closed sets. Specifically, for disjoint closed sets A and B, there exists a function U assigning to each such pair an open set U(A, B) containing A and disjoint from B, such that if A' \subseteq A and B' \subseteq B are disjoint closed sets, then 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.^[24] Collectionwise normal spaces strengthen normality by handling families of closed sets simultaneously. A T_1 space is collectionwise normal if, for every discrete collection \{F_i\}_{i \in I} of closed sets, there exists a discrete collection \{U_i\}_{i \in I} of open sets such that F_i \subseteq U_i for each i and the U_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.^[25] Completely normal spaces that are also Lindelöf are paracompact, as the Lindelöf property combined with normality (which complete normality implies under T_1) yields a locally finite open refinement for every open cover via standard theorems on regular Lindelöf spaces.^[26]

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