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Metrizable space
Metrizable space
Metrizable space
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In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is [1][2] Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.

Properties

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Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic.

Metrization theorems

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One of the first widely recognized metrization theorems was Urysohn's metrization theorem. This states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable. (Historical note: The form of the theorem shown here was in fact proved by Tikhonov in 1926. What Urysohn had shown, in a paper published posthumously in 1925, was that every second-countable normal Hausdorff space is metrizable.) The converse does not hold: there exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric.[3] The Nagata–Smirnov metrization theorem, described below, provides a more specific theorem where the converse does hold.

Several other metrization theorems follow as simple corollaries to Urysohn's theorem. For example, a compact Hausdorff space is metrizable if and only if it is second-countable.

Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable. The Nagata–Smirnov metrization theorem extends this to the non-separable case. It states that a topological space is metrizable if and only if it is regular, Hausdorff and has a σ-locally finite base. A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets. For a closely related theorem see the Bing metrization theorem.

Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube that is, the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the product topology.

A space is said to be locally metrizable if every point has a metrizable neighbourhood. Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff and paracompact. In particular, a manifold is metrizable if and only if it is paracompact.

Examples

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The group of unitary operators on a separable Hilbert space endowed with the strong operator topology is metrizable (see Proposition II.1 in [4]).

Non-normal spaces cannot be metrizable; important examples include

The real line with the lower limit topology is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff, paracompact and first countable.

Locally metrizable but not metrizable

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The Line with two origins, also called the bug-eyed line is a non-Hausdorff manifold (and thus cannot be metrizable). Like all manifolds, it is locally homeomorphic to Euclidean space and thus locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff). It is also a T1 locally regular space but not a semiregular space.

The long line is locally metrizable but not metrizable; in a sense, it is "too long".

See also

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References

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Metrizable space

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In topology, a metrizable space is a topological space (X,τ)(X, \tau) for which there exists a metric dd on the set XX such that the topology τ\tau is precisely the topology induced by dd, meaning that the open sets in τ\tau are exactly the unions of open balls defined by dd. This compatibility allows metrizable spaces to inherit many useful properties from metric spaces, such as the ability to define distances and Cauchy sequences (with completeness depending on the metric), although the specific metric may not be unique and different metrics can induce the same topology. Metrizability is a topological invariant, preserved under homeomorphisms, so if one space is homeomorphic to a metrizable space, it is itself metrizable. A fundamental result characterizing metrizable spaces is the Urysohn metrization theorem, which states that every second-countable regular Hausdorff topological space is metrizable. Here, second-countability refers to the existence of a countable basis for the topology, and regularity means that for every point and closed set not containing it, there are disjoint open neighborhoods separating them. This theorem embeds such spaces into the Hilbert cube [0,1]N[0,1]^\mathbb{N}, a complete metric space, highlighting their "metric-like" structure. Metrizable spaces exhibit strong separation axioms, including normality (T4), which enables the for continuous functions, and paracompactness, ensuring the existence of locally finite refinements of open covers. They are also completely regular, allowing continuous functions to separate points from closed sets. These properties make metrizable spaces central in and , as they bridge abstract with concrete metric tools like convergence and criteria.

Fundamentals

Definition

A topological space (X,τ)(X, \tau) is metrizable if there exists a metric dd on XX such that the τ\tau is the one generated by the open balls B(x,r)={y∈X∣d(x,y)<r}B(x, r) = \{y \in X \mid d(x, y) < r\} for x∈Xx \in X and r>0r > 0. A metric d:X×X→[0,∞)d: X \times X \to [0, \infty) on a set XX satisfies the following axioms: d(x,y)≥0d(x, y) \geq 0 for all x,y∈Xx, y \in X, with d(x,y)=0d(x, y) = 0 if and only if x=yx = y; d(x,y)=d(y,x)d(x, y) = d(y, x) for all x,y∈Xx, y \in X; and d(x,z)≤d(x,y)+d(y,z)d(x, z) \leq d(x, y) + d(y, z) for all x,y,z∈Xx, y, z \in X (the triangle inequality). The induced by the metric consists of all unions of such open balls, where a set U⊆XU \subseteq X is open if for every x∈Ux \in U, there exists r>0r > 0 such that B(x,r)⊆UB(x, r) \subseteq U. A metrizable space is thus homeomorphic to a , meaning it admits a compatible with some metric, though the metric inducing the topology is not necessarily unique, as different metrics can generate the same topology. A basic example is the real line R\mathbb{R} with the standard topology, induced by the Euclidean metric d(x,y)=∣x−y∣d(x, y) = |x - y|. Every metrizable space is Hausdorff.

Historical Development

The concept of metrizable spaces emerged from early 20th-century efforts to abstract and generalize notions of distance and convergence beyond . In 1906, Maurice Fréchet introduced the idea of abstract metric spaces in his doctoral dissertation, providing a framework for spaces equipped with a distance function that satisfied basic axioms, laying the groundwork for later topological developments. This work extended classical analysis to arbitrary sets, influencing the study of continuity and limits in non-traditional settings. Felix Hausdorff's 1914 book Grundzüge der Mengenlehre marked a pivotal advancement by defining topological spaces axiomatically through neighborhoods, without relying on metrics, which separated from metric-based approaches and set the stage for investigating when such spaces could be metrized. Pavel Urysohn built on this in 1925 with his metrization theorem, proving that second-countable regular Hausdorff spaces admit a compatible metric; tragically, Urysohn drowned at age 26 in 1924, before seeing the full publication of his results in Mathematische Annalen. contributed in 1934 by refining axiomatic foundations of topology, including early ideas on uniform structures that bridged metric and topological properties. Subsequent milestones addressed broader cases beyond second-countability. In 1951, R.H. Bing provided a characterization of metrizable spaces as collectionwise normal Moore spaces in his paper in the Canadian Journal of Mathematics. Independently, around the same time, Jun-Ichi Nagata and Yu. M. Smirnov developed conditions involving σ-locally finite bases for regular Hausdorff spaces, culminating in the Nagata-Smirnov metrization theorem (published in 1950 and 1951, respectively), which generalized Urysohn's result to non-second-countable settings. These developments from the to shifted focus from concrete Euclidean metrics to abstract topologies, profoundly impacting by enabling metric-like tools in infinite-dimensional contexts.

Characterizations

Topological Characterizations

A key topological characterization of metrizable spaces is provided by Urysohn's metrization theorem, which states that every regular possessing a second-countable basis is metrizable. This condition ensures the existence of a metric compatible with the without explicitly constructing one, relying instead on the interplay between regularity, Hausdorff separation, and the countability of the basis. The theorem highlights that second countability imposes a strong form of "tameness" on the , allowing for effective control over open covers and separations. Weaker topological conditions fail to guarantee metrizability. For instance, the Niemytzki plane (also known as the Moore plane or tangent disc space) is a regular that is developable but not metrizable, serving as a to the sufficiency of regularity and developability alone. In this space, the upper half-plane is equipped with the standard while the x-axis receives a finer topology generated by tangent discs, resulting in a non-normal subspace that prevents metrizability despite the overall regularity. Several equivalent topological conditions capture metrizability precisely. A space is metrizable if and only if it is a regular Hausdorff (T3) space with a second-countable basis. Equivalently, it is T3 and Lindelöf with a σ-locally finite basis, though the Lindelöf property alone with T3 does not suffice without additional structure like local metrizability. Another formulation involves developable spaces: a collectionwise normal developable space (a Moore space satisfying collectionwise normality) is metrizable, as established by Bing's metrization criterion. The essential role of countability in these characterizations stems from the second-countable basis, which permits the enumeration of open sets to construct separating functions and covers. This countability facilitates embedding the space into a product of intervals via continuous Urysohn functions, one for each basis element, yielding a homeomorphism into the Hilbert cube [0,1]N[0,1]^\mathbb{N}, a compact metrizable space. Such embeddings underscore how topological countability axioms enable metric realization without invoking uniform structures.

Uniform Characterizations

A uniform structure on a set XX provides a framework for generalizing notions of continuity and completeness beyond metric spaces, allowing the definition of uniform continuity and Cauchy sequences in a topological context. Introduced by André Weil in 1937, uniform structures bridge metrics and topologies by abstracting the idea of "nearness" between points without relying on a specific distance function. In this perspective, a topological space is metrizable if and only if it admits a compatible uniform structure that is induced by some metric on the space. The uniform structure is defined through a collection of subsets of X×XX \times X called entourages, which satisfy specific axioms: reflexivity (the diagonal Δ={(x,x)∣x∈X}\Delta = \{(x,x) \mid x \in X\} is contained in every entourage), symmetry (if EE is an entourage, so is its transpose ET={(y,x)∣(x,y)∈E}E^T = \{(y,x) \mid (x,y) \in E\}), and the triangle inequality (for every entourage EE, there exists an entourage FF such that F∘F⊆EF \circ F \subseteq E, where ∘\circ denotes ). A base for the uniformity is a filter subbase consisting of entourages such that every entourage contains one from the base. A metric dd on XX induces a uniformity whose base consists of the open balls' entourages Eε={(x,y)∈X×X∣d(x,y)<ε}E_\varepsilon = \{(x,y) \in X \times X \mid d(x,y) < \varepsilon\} for ε>0\varepsilon > 0, satisfying the entourage axioms since the metric ensures symmetry and the . A key characterization states that a Hausdorff is metrizable if and only if its uniformity admits a countable base of entourages. This condition ensures the existence of a compatible metric generating the uniformity, as the countable base allows of a single metric via a suitable and , akin to into a product of pseudometric spaces. For topological spaces, this translates to metrizability when the space is completely regular (uniformizable in a Hausdorff manner) and the induced uniformity has a countable entourage base, providing an extension of purely topological criteria by incorporating quantitative uniformity aspects. In metrizable uniform spaces, the entourage structure enables the definition of Cauchy sequences: a sequence (xn)(x_n) is Cauchy if for every entourage EE, there exists NN such that (xn,xm)∈E(x_n, x_m) \in E for all n,m≥Nn, m \geq N. However, completeness requires a stronger condition; while every complete metric induces a complete uniformity (where every Cauchy filter converges), a metrizable uniformity is complete only if it arises from a complete metric, distinguishing intrinsic completeness from mere metrizability. This framework highlights how uniform structures generalize metric properties, facilitating analysis in non-metric settings while preserving essential features like .

Properties

Separation and Countability Properties

Metrizable spaces satisfy a range of separation axioms, making them highly structured . Specifically, they are T1T_1 (with singleton sets closed), T2T_2 (Hausdorff, as distinct points can be separated by disjoint open balls), T3T_3 (regular Hausdorff, allowing separation of points from closed sets not containing them via distance-based open sets), T4T_4 (normal Hausdorff, enabling separation of disjoint closed sets by disjoint open sets using Urysohn-type functions derived from metrics), and Tychonoff (completely regular Hausdorff, where points can be separated from closed sets by continuous functions to [0,1][0,1], such as the distance function normalized appropriately). These properties follow directly from the metric inducing the topology, ensuring robust separation capabilities without additional assumptions. A fundamental countability property of metrizable spaces is first-countability: at each point xx, there exists a countable local basis, such as the sequence of open balls B(x,1/n)B(x, 1/n) for n∈Nn \in \mathbb{N}. This local countability facilitates the use of sequences to probe topological features, such as convergence and . In particular, first-countability ensures that sequential convergence coincides with topological convergence; a {xn}\{x_n\} converges to xx if and only if every neighborhood of xx contains all but finitely many xnx_n, mirroring the filter-based definition. Consequently, in metrizable spaces, continuous functions preserve sequential limits, providing a sequential criterion for continuity that simplifies analysis compared to general topological spaces. Regarding global countability, a metrizable space is second-countable (possessing a countable basis for the ) if and only if it is separable (admitting a countable dense ). To see the equivalence, separability allows construction of a countable basis from rational-radius balls centered at the dense points, while second-countability implies separability via selecting points from basis elements. This interplay also ties into characterizations: in metrizable spaces, sequential compactness (every has a convergent ) is equivalent to due to first-countability. Metrizability does not, however, imply the Lindelöf property (every open cover has a countable subcover) in general. For instance, an equipped with the discrete metric is metrizable but not Lindelöf, as the open cover by singletons requires an uncountable subcover. Nonetheless, if a metrizable space is additionally Lindelöf, it must be second-countable (and hence separable), establishing a bridge between these countability conditions under the extra assumption.

Paracompactness and Uniformity

A metrizable space, being Hausdorff, is paracompact: every open cover admits a locally finite open refinement. This property, established by Stone, ensures that metrizable spaces satisfy a strengthening of compactness in terms of cover refinements, facilitating the construction of structures like partitions of unity. Specifically, in a paracompact Hausdorff space, for any open cover, there exists a partition of unity subordinate to it—continuous functions summing to 1 with supports contained in the cover sets—which is particularly useful in differential topology for gluing local constructions globally. Metrizable spaces are uniformizable: the metric induces a compatible uniform structure via entourages of the form {(x,y)∣d(x,y)<ϵ}\{(x,y) \mid d(x,y) < \epsilon\} for ϵ>0\epsilon > 0, generating the uniformity whose induced topology is the original one. This uniformity is translation-invariant in the Hausdorff case and allows metrizable spaces to be embedded into larger uniform structures while preserving topological properties. Since metrizable spaces are completely regular, they admit a Stone-Čech compactification βX\beta X, the unique (up to homeomorphism) compact Hausdorff space containing XX as a dense subspace such that every bounded continuous real-valued function on XX extends continuously to βX\beta X. This compactification relates to the Tychonoff theorem, as βX\beta X can be realized as the closure of XX in the product of intervals [0,1]I[0,1]^I over all continuous functions from XX to [0,1][0,1], embedding XX into a Tychonoff product. Moreover, βX\beta X is metrizable if and only if XX is compact. Paracompactness distinguishes metrizable spaces from certain non-metrizable manifolds; for instance, the long line, a connected 1-dimensional Hausdorff manifold that is locally Euclidean but not second countable, fails to be paracompact as some open covers lack locally finite refinements.

Metrization Theorems

Urysohn's Metrization Theorem

Urysohn's metrization theorem states that every second-countable regular Hausdorff space is metrizable. This result provides a fundamental characterization of metrizable spaces among those with a countable basis. The theorem was proved by Pavel Urysohn in 1925, building on his earlier work in point-set topology during a 1924 European trip where he collaborated with Pavel Aleksandrov. The proof relies on the regularity of the space to apply , which guarantees the existence of continuous functions separating points from closed sets. To sketch the proof, consider a second-countable regular Hausdorff space (X,τ)(X, \tau) with countable basis {Bn:n∈N}\{B_n : n \in \mathbb{N}\}. First, normality follows from second countability and regularity, allowing the construction of continuous functions fm,n:X→[0,1]f_{m,n}: X \to [0,1] via to separate the closed sets X∖BnX \setminus B_n and the closure of BmB_m for each pair (m,n)(m,n). Enumerate these functions as a countable family {fk:k∈N}\{f_k : k \in \mathbb{N}\} that separates points, yielding a continuous injection F:X→[0,1]NF: X \to [0,1]^\mathbb{N} defined by F(x)=(f1(x),f2(x),… )F(x) = (f_1(x), f_2(x), \dots). This embeds XX homeomorphically into the [0,1]N[0,1]^\mathbb{N}, which is metrizable with metric d(x,y)=∑n=1∞12n∣fn(x)−fn(y)∣.d(x,y) = \sum_{n=1}^\infty \frac{1}{2^n} |f_n(x) - f_n(y)|. The series converges since each term is at most 1/2n1/2^n, and dd induces the on the image, confirming metrizability. The theorem's reliance on second countability imposes limitations; for instance, the long line, a regular that is locally metrizable but not second-countable, fails to be metrizable.

Nagata–Smirnov Metrization Theorem

The Nagata–Smirnov metrization theorem provides a characterization of metrizable spaces that generalizes Urysohn's theorem to spaces without second countability, addressing the metrization problem for a broader class of topological spaces. Independently proved by Jun-iti Nagata in and Yuriĭ Smirnov in , the theorem resolves cases where the base is uncountable by imposing a countable structure on locally finite refinements. This result marked a significant advancement in , building on earlier work while handling non-second-countable regular s. The states that a XX is metrizable if and only if it is regular and has a , where a σ\sigma-locally finite basis is a basis B\mathcal{B} for the of XX that can be expressed as a countable union B=⋃n=1∞Bn\mathcal{B} = \bigcup_{n=1}^\infty \mathcal{B}_n such that each Bn\mathcal{B}_n is locally finite (every point in XX has a neighborhood intersecting only finitely many members of Bn\mathcal{B}_n). Regularity here means that for every point x∈Xx \in X and C⊆XC \subseteq X with x∉Cx \notin C, there exist disjoint open sets separating xx from CC. Second countability is a special case of this condition, as a countable basis is σ\sigma-locally finite with each Bn\mathcal{B}_n finite. The proof of sufficiency constructs a metric on XX by it into a product of intervals with the metric. Given a σ\sigma-locally finite basis {Bn}n∈N\{\mathcal{B}_n\}_{n \in \mathbb{N}}, for each nn and B∈BnB \in \mathcal{B}_n, define a fn,B:X→[0,1/n]f_{n,B}: X \to [0, 1/n] that is zero outside BB and positive inside, using regularity to separate points via Urysohn-type functions scaled by diameters: assign sets in Bn\mathcal{B}_n diameters at most 2/n2/n to ensure the embedding preserves distances appropriately. The σ\sigma-condition guarantees the product space [0,1]J[0,1]^J (over the JJ of such functions) admits a metric inducing the on the image of XX, which is homeomorphic to XX. Necessity follows from the fact that any metrizable space is regular and admits a σ\sigma-locally finite basis obtained by refining open covers of balls of 1/n1/n into locally finite families with controlled diameters. A regular space satisfying the σ\sigma-locally finite basis condition is paracompact, as the basis allows refinements of open covers into locally finite ones via countable unions, and regularity ensures the existence of partitions of unity subordinate to such refinements by Stone's theorem. This links the theorem to paracompactness characterizations of metrizability.

Examples

Standard Metrizable Spaces

Euclidean spaces provide fundamental examples of metrizable spaces. The space Rn\mathbb{R}^n equipped with the Euclidean metric d(x,y)=∑i=1n(xi−yi)2d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}
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