Metrizable space
Metrizable space
Main page
M

Metrizable space

logo
Community Hub0 subscribers
Read side by side
Metrizable space
View on Wikipedia
from Wikipedia

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

[edit]

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

[edit]

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

[edit]

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

[edit]

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

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Metrizable space

View on Grokipedia
from Grokipedia
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.[1] 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.[2] Metrizability is a topological invariant, preserved under homeomorphisms, so if one space is homeomorphic to a metrizable space, it is itself metrizable.[1] A fundamental result characterizing metrizable spaces is the Urysohn metrization theorem, which states that every second-countable regular Hausdorff topological space is metrizable.[3] 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.[4] 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.[5] Metrizable spaces exhibit strong separation axioms, including normality (T4), which enables the Tietze extension theorem 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.[6] These properties make metrizable spaces central in analysis and geometry, as they bridge abstract topology with concrete metric tools like convergence and compactness criteria.[7]

Fundamentals

Definition

A topological space (X,τ)(X, \tau) is metrizable if there exists a metric dd on XX such that the topology 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.[8][1] 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).[9] The topology 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.[10][11] A metrizable space is thus homeomorphic to a metric space, meaning it admits a topology compatible with some metric, though the metric inducing the topology is not necessarily unique, as different metrics can generate the same topology.[8] 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|.[12] Every metrizable space is Hausdorff.[13]

Historical Development

The concept of metrizable spaces emerged from early 20th-century efforts to abstract and generalize notions of distance and convergence beyond Euclidean geometry. 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.[14] 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 general topology from metric-based approaches and set the stage for investigating when such spaces could be metrized.[15] 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.[16][17] Andrey Kolmogorov contributed in 1934 by refining axiomatic foundations of topology, including early ideas on uniform structures that bridged metric and topological properties.[15] 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.[18] 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.[17] These developments from the 1920s to 1950s shifted focus from concrete Euclidean metrics to abstract topologies, profoundly impacting functional analysis by enabling metric-like tools in infinite-dimensional contexts.[15]

Characterizations

Topological Characterizations

A key topological characterization of metrizable spaces is provided by Urysohn's metrization theorem, which states that every regular Hausdorff space possessing a second-countable basis is metrizable.[19] This condition ensures the existence of a metric compatible with the topology 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 topology, 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 Hausdorff space that is developable but not metrizable, serving as a counterexample to the sufficiency of regularity and developability alone.[20] In this space, the upper half-plane is equipped with the standard topology 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.[21] 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.[22] 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.[23] 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.[21] 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.[24] 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 composition of relations).[25] 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 triangle inequality.[24] A key characterization states that a Hausdorff uniform space 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 construction of a single metric via a suitable enumeration and summation, akin to embedding into a product of pseudometric spaces.[26] 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.[24] 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.[25] This framework highlights how uniform structures generalize metric properties, facilitating analysis in non-metric settings while preserving essential features like uniform continuity.[24]

Properties

Separation and Countability Properties

Metrizable spaces satisfy a range of separation axioms, making them highly structured topologically. 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).[22] 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}.[22] This local countability facilitates the use of sequences to probe topological features, such as convergence and compactness. In particular, first-countability ensures that sequential convergence coincides with topological convergence; a sequence {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.[27] 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 topology) if and only if it is separable (admitting a countable dense subset).[28] 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 compactness characterizations: in metrizable spaces, sequential compactness (every sequence has a convergent subsequence) is equivalent to compactness due to first-countability.[27] Metrizability does not, however, imply the Lindelöf property (every open cover has a countable subcover) in general. For instance, an uncountable set equipped with the discrete metric is metrizable but not Lindelöf, as the open cover by singletons requires an uncountable subcover.[1] 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.[29]

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.[30] 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.[31] 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.[31] Moreover, βX\beta X is metrizable if and only if XX is compact.[32] 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.[33]

Metrization Theorems

Urysohn's Metrization Theorem

Urysohn's metrization theorem states that every second-countable regular Hausdorff space is metrizable.[3] This result provides a fundamental characterization of metrizable spaces among those with a countable basis.[21] 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.[16] The proof relies on the regularity of the space to apply Urysohn's lemma, which guarantees the existence of continuous functions separating points from closed sets.[21] 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 Urysohn's lemma to separate the closed sets X∖BnX \setminus B_n and the closure of BmB_m for each pair (m,n)(m,n).[21] 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).[3] This embeds XX homeomorphically into the Hilbert cube [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 product topology on the image, confirming metrizability.[21] The theorem's reliance on second countability imposes limitations; for instance, the long line, a regular Hausdorff space that is locally metrizable but not second-countable, fails to be metrizable.[34]

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 1950 and Yuriĭ Smirnov in 1951, the theorem resolves cases where the base is uncountable by imposing a countable structure on locally finite refinements.[35] This result marked a significant advancement in general topology, building on earlier work while handling non-second-countable regular Hausdorff spaces.[36] The theorem states that a topological space XX is metrizable if and only if it is regular and has a σ\sigma-locally finite basis, where a σ\sigma-locally finite basis is a basis B\mathcal{B} for the topology 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).[35] Regularity here means that for every point x∈Xx \in X and closed set C⊆XC \subseteq X with x∉Cx \notin C, there exist disjoint open sets separating xx from CC.[37] 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.[36] The proof of sufficiency constructs a metric on XX by embedding it into a product of intervals with the uniform 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 continuous function 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 index set JJ of such functions) admits a uniform metric inducing the topology on the image of XX, which is homeomorphic to XX.[37] 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 radius 1/n1/n into locally finite families with controlled diameters.[36] 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.[37] This links the theorem to paracompactness characterizations of metrizability.[36]

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} induces the standard topology, which is Hausdorff, second-countable, and locally compact.[22] This metric ensures that open balls form a basis for the topology, making Rn\mathbb{R}^n a complete and separable metric space for each finite dimension nn.[38] Hilbert spaces, such as the sequence space ℓ2\ell^2 of square-summable real sequences, are also standard metrizable spaces. The metric derived from the inner product norm ∥x∥2=∑n=1∞∣xn∣2\|x\|_2 = \sqrt{\sum_{n=1}^\infty |x_n|^2} renders ℓ2\ell^2 a complete and separable metric space, with the rational polynomials forming a countable dense subset.[39] This completeness and separability highlight ℓ2\ell^2 as a prototypical infinite-dimensional Hilbert space in functional analysis.[40] Discrete spaces exemplify metrizability in a simple, zero-dimensional context. For any set XX, the discrete metric d(x,y)=1d(x, y) = 1 if x≠yx \neq y and d(x,x)=0d(x, x) = 0 induces the discrete topology, where every subset is open and the space is zero-dimensional with clopen singletons as a basis.[41] This metric satisfies the triangle inequality and makes any discrete space metrizable, regardless of cardinality, though separability holds only for countable XX.[42] Smooth manifolds are metrizable through metrics induced by their atlas of charts. A smooth nn-manifold is locally Euclidean, with the topology generated by charts to open subsets of Rn\mathbb{R}^n, and paracompactness ensures a compatible Riemannian metric that induces the manifold topology globally.[43] This metrizability follows from the second-countability and Hausdorff properties inherent in the definition of smooth manifolds.[44] The space of continuous functions C([0,1])C([0,1]) on the unit interval, equipped with the supremum metric d(f,g)=sup⁡x∈[0,1]∣f(x)−g(x)∣d(f, g) = \sup_{x \in [0,1]} |f(x) - g(x)|, is a classic metrizable function space. This metric induces a complete and separable topology, making C([0,1])C([0,1]) a Polish space, with the countable set of polynomials with rational coefficients dense in it.[45] Such spaces are central in analysis for studying uniform convergence and approximation.[46] These examples illustrate second-countability, a key property linking metrizability to countability axioms, as detailed in the section on separation and countability properties.

Non-Metrizable Counterexamples

The Sorgenfrey line, denoted Rl\mathbb{R}_l, is the set of real numbers R\mathbb{R} endowed with the lower limit topology, whose basis consists of all half-open intervals [a,b)[a, b) for a<ba < b in R\mathbb{R}. This topology renders Rl\mathbb{R}_l Hausdorff, regular, and normal, with the rational numbers forming a dense subset, making it separable. However, Rl\mathbb{R}_l lacks a countable basis. To see this, note that for each real number b∈Rb \in \mathbb{R}, any basis element VbV_b containing bb with inf⁡Vb=b\inf V_b = b must be distinct for different bb, as the infimum determines the starting point uniquely; thus, the basis must have at least continuum cardinality, which is uncountable.[47] Consequently, Rl\mathbb{R}_l violates the second-countability condition of Urysohn's metrization theorem and thus admits no compatible metric. Additionally, its basis fails to be a countable union of locally finite families, contravening the Nagata–Smirnov metrization theorem.[48] Another counterexample arises from the one-point compactification of an uncountable discrete space. Let DD be an uncountable set with the discrete topology, where every subset is open. The one-point compactification αD\alpha D adjoins a point ∞\infty to DD, with open sets comprising the open subsets of DD and complements in αD\alpha D of finite subsets of DD. This construction yields a compact Hausdorff space, as finite subsets of DD are the only compact sets in DD.[49] Yet αD\alpha D is not second-countable: the points of DD constitute an uncountable closed discrete subspace, since singletons remain open and separated.[49] Compact metrizable spaces are necessarily second-countable, so αD\alpha D cannot be metrizable, again violating Urysohn's theorem.[49] The basis for αD\alpha D, including all singletons of DD and neighborhoods of ∞\infty, cannot be expressed as a countable union of locally finite families, breaching the Nagata–Smirnov condition.[49][48] The deleted Tychonoff plank provides a further illustration of non-metrizability through separation failure. Define the Tychonoff plank as the product [0,ω1]×[0,ω][0, \omega_1] \times [0, \omega], where ω1\omega_1 is the first uncountable ordinal and ω\omega the first countably infinite ordinal, both with the order topology; the deleted version removes the corner point (ω1,ω)(\omega_1, \omega). This space is locally compact and Hausdorff, inheriting these properties from the compact product space.[50] Nevertheless, it is not normal: the sets A={ω1}×[0,ω)A = \{\omega_1\} \times [0, \omega) and B=[0,ω1]×{ω}B = [0, \omega_1] \times \{\omega\} are closed and disjoint, but no disjoint open sets separate them, as any neighborhood of AA extends toward the deleted point and intersects any neighborhood of BB.[50] Lacking normality—a property satisfied by all metrizable spaces—despite being regular, the deleted Tychonoff plank is non-metrizable. Its basis, derived from the product, is neither second-countable nor σ\sigma-locally finite due to the uncountable ordinal factor.[50]

Locally Metrizable but Not Globally Metrizable Spaces

A topological space is locally metrizable if for every point in the space, there exists an open neighborhood that is metrizable when equipped with the subspace topology.[51] This property ensures that the space behaves like a metric space in sufficiently small regions around each point, but it does not guarantee the existence of a single metric compatible with the entire topology. One classic example is the line with double origin, constructed by taking two copies of the real line R\mathbb{R} and identifying all points except the origins, resulting in a space with two distinct origin points, denoted 010_1 and 020_2. The topology is generated by the usual open intervals on each copy, with basis elements around non-origin points being standard open intervals and around each origin being intervals that include only that origin and symmetric points away from it. This space is locally homeomorphic to R\mathbb{R} at every point, hence locally metrizable, as small neighborhoods around any point, including the origins, are homeomorphic to open intervals in R\mathbb{R}, which are metrizable. However, it is not globally metrizable because it fails to be Hausdorff: the two origins cannot be separated by disjoint open sets, as any neighborhood of one intersects any neighborhood of the other at points sufficiently close to zero.[51] Another prominent example is the long line, formed by taking the first uncountable ordinal ω1\omega_1 and replacing each point α<ω1\alpha < \omega_1 with a half-open interval [0,1)[0, 1) under the lexicographic order topology, then adjoining a point at the end to form the extended long line if desired. Locally, the space is homeomorphic to R\mathbb{R}, making it a one-dimensional manifold without boundary and thus locally metrizable. Despite this, the long line is not globally metrizable, as it is not second-countable—its topology has uncountably many disjoint open intervals corresponding to the uncountable ordinals, violating the countability condition required by Urysohn's metrization theorem for regular Hausdorff spaces.[52] These examples illustrate that local metrizability is strictly weaker than global metrizability, as the former holds in regions but fails to extend compatibly across the space due to global pathologies like non-Hausdorff separation or lack of second countability. In particular, while locally metrizable spaces inherit local properties such as first-countability from metric neighborhoods, global metrizability demands additional conditions like those in the metrization theorems.[51][52]

Related Concepts

Uniform Spaces

Uniform spaces generalize the structure of metric spaces by abstracting the notion of "nearness" through a collection of binary relations known as entourages, rather than a specific distance function. A uniform structure on a set XX is a filter U\mathcal{U} on the power set of X×XX \times X consisting of entourages that satisfy three key properties: each entourage U∈UU \in \mathcal{U} contains the diagonal ΔX={(x,x)∣x∈X}\Delta_X = \{(x, x) \mid x \in X\}; the filter is symmetric, meaning if U∈UU \in \mathcal{U}, then U−1={(y,x)∣(x,y)∈U}∈UU^{-1} = \{(y, x) \mid (x, y) \in U\} \in \mathcal{U}; and it is transitive in the sense that if U∈UU \in \mathcal{U}, then there exists V∈UV \in \mathcal{U} such that V∘V⊆UV \circ V \subseteq U, where ∘\circ denotes relational composition (V∘V)={(x,z)∣∃y∈X:(x,y)∈V,(y,z)∈V}(V \circ V) = \{(x, z) \mid \exists y \in X : (x, y) \in V, (y, z) \in V\}. This framework allows the definition of uniform continuity, Cauchy sequences (or more generally, filters), and completeness without relying on metrics. Every metric space (X,d)(X, d) induces a uniform structure on XX via the basis of entourages {Uϵ∣ϵ>0}\{U_\epsilon \mid \epsilon > 0\}, where Uϵ={(x,y)∈X×X∣d(x,y)<ϵ}U_\epsilon = \{(x, y) \in X \times X \mid d(x, y) < \epsilon\}, demonstrating how uniform spaces encompass metric spaces as a special case. A key connection between uniform spaces and metrizability arises through a precise criterion: a uniform space (X,U)(X, \mathcal{U}) is metrizable—meaning U\mathcal{U} is induced by some metric on XX—if and only if the topology induced by U\mathcal{U} is Hausdorff and U\mathcal{U} admits a countable basis {Un∣n∈N}\{U_n \mid n \in \mathbb{N}\}. The Hausdorff condition ensures separation of distinct points via entourages that avoid off-diagonal pairs, while the countable basis guarantees the uniformity can be "tamed" by a sequence of shrinking neighborhoods analogous to balls in a metric. This criterion highlights how metrizable uniform spaces recover the full structure of metric spaces, including the ability to define distances explicitly from the entourages. Not all uniform spaces are metrizable, providing examples that illustrate the broader applicability of the concept. The indiscrete uniformity on a set XX with ∣X∣>1|X| > 1, generated by the single entourage X×XX \times X, forms a valid uniform structure but induces the indiscrete topology, which is not Hausdorff and thus not metrizable. Similarly, non-Hausdorff uniform spaces, such as those arising from quotient constructions where points cannot be separated, fail metrizability despite satisfying the entourage axioms. Another prominent example is the product uniform space ∏i∈I[0,1]\prod_{i \in I} [0,1] for an uncountable index set II, equipped with the product uniformity (each factor metrizable via the standard metric); this space lacks a countable entourage basis due to the uncountable product, rendering it non-metrizable even though its induced topology is compact and Hausdorff. These cases underscore the necessity of the countable basis and Hausdorff conditions for metrizability. Uniform spaces find significant applications in extending metric concepts to structures like topological groups, where no canonical metric exists but uniformity can still be defined. A topological group GG carries natural left and right uniform structures: the left uniformity has basis entourages UL={(g,h)∣g−1h∈V}U_L = \{(g, h) \mid g^{-1}h \in V\} for symmetric neighborhoods VV of the identity, and the right uniformity is defined analogously via hg−1∈Vh g^{-1} \in V. Metrizable uniformities on such groups, satisfying the aforementioned criterion, allow the formulation of Cauchy filters—generalizations of Cauchy sequences defined as proper filters F\mathcal{F} on XX such that for every entourage U∈UU \in \mathcal{U}, there exists F∈FF \in \mathcal{F} with F×F⊆UF \times F \subseteq U—enabling constructions like completions and uniform continuity in non-metric settings. For instance, in metrizable topological groups, every Cauchy filter converges, facilitating the study of completeness and convergence in abstract algebraic topologies. The relationship between uniform and topological spaces is mediated by the forgetful functor from the category Unif of uniform spaces to the category Top of topological spaces, which assigns to each uniform space (X,U)(X, \mathcal{U}) its induced topology τU\tau_{\mathcal{U}}, generated by sets W⊆XW \subseteq X such that for every x∈Wx \in W, there exists U∈UU \in \mathcal{U} with (x,y)∈U(x, y) \in U implying y∈Wy \in W. This functor preserves metrizability under compatible conditions: a uniform space is metrizable if and only if its underlying topological space is metrizable, meaning the induced topology admits a compatible uniformity generated by a metric. Thus, uniform spaces provide a categorical bridge, embedding metric properties into the broader topological framework while ensuring that metrizable instances align precisely with metrizable topologies.

Fréchet Spaces and Metrizability

In functional analysis, a Fréchet space is defined as a complete, Hausdorff, locally convex topological vector space whose topology is induced by a countable family of seminorms, rendering it metrizable with a translation-invariant metric.[53] This metrizability ensures the space admits a compatible metric that respects the vector space operations, facilitating the application of analytical tools such as fixed-point theorems.[53] Specifically, the completeness with respect to this metric allows the Banach fixed-point theorem to apply to contractions on the space, enabling the study of existence and uniqueness for operator equations in such settings.[54] Prominent examples include the Schwartz space of rapidly decreasing smooth functions on Rn\mathbb{R}^n, which serves as the space of test functions in distribution theory and is a nuclear Fréchet space equipped with seminorms involving suprema over derivatives and polynomial weights.[53] Similarly, Sobolev spaces Hk(Rn)H^k(\mathbb{R}^n) for fixed integer k≥0k \geq 0, defined via weak derivatives in L2L^2, are Hilbert spaces and thus Fréchet spaces, providing a framework for embedding theorems and regularity results in analysis.[55] These spaces highlight the role of metrizability in ensuring sequential completeness, which is essential for convergence properties in infinite-dimensional settings. Not all locally convex completions of inductive limits qualify as Fréchet spaces; for instance, LF-spaces formed as strict inductive limits of sequences of Fréchet spaces, such as the space of compactly supported smooth functions D(Ω)\mathcal{D}(\Omega), are typically not metrizable due to their non-countable neighborhood bases.[56] This distinction underscores the necessity of metrizability for Fréchet structures, as non-metrizable LF-spaces lack a compatible translation-invariant metric despite their completeness in a uniform sense. Fréchet spaces have found significant applications in 20th- and 21st-century mathematics, particularly in partial differential equations (PDEs), where semigroup theory in these spaces resolves evolution equations and boundary-value problems.[57] In quantum field theory, the completeness and metrizability of spaces like the Schwartz space enable rigorous treatments of test functions and distributions, supporting the algebraic and analytic foundations for field observables on smooth sections of bundles.

References

  1. [PDF] Metrizable spaces
  2. [PDF] 12. Metric spaces and metrizability
  3. [PDF] 15. Urysohn's metrization theorem
  4. [PDF] integration workshop project: the urysohn metrization theorem
  5. [PDF] 12 | Urysohn Metrization Theorem
  6. [PDF] Chapter 6. Metrization Theorems and Paracompactness
  7. [PDF] an exploration of the metrizability of topological spaces
  8. https://www.math.toronto.edu/ivan/mat327/docs/notes/12-metric.pdf
  9. [PDF] Section 20. The Metric Topology
  10. [PDF] Chapter 9 The Topology of Metric Spaces - UCI Mathematics
  11. [PDF] Metric Topology I - Harvard Mathematics Department
  12. [PDF] Chapter 13: Metric, Normed, and Topological Spaces - UC Davis Math
  13. [PDF] the intersection of topological and metric spaces - UChicago Math
  14. Maurice Fréchet (1878 - 1973) - Biography - MacTutor
  15. A history of Topology - MacTutor - University of St Andrews
  16. Pavel Urysohn (1898 - 1924) - Biography - MacTutor
  17. A History of Generalized Metrizable Spaces - SpringerLink
  18. Metrization of Topological Spaces | Canadian Journal of Mathematics
  19. Urysohn metrization theorem - Encyclopedia of Mathematics
  20. S74: Niemytzki plane - π-Base
  21. Urysohn metrization theorem in nLab
  22. [PDF] 12. Metric spaces and metrizability
  23. 2. A Note on the Metrizability o M.Spaces - Project Euclid
  24. General Topology - John L. Kelley - Google Books
  25. uniform space in nLab
  26. [PDF] Notes on Uniform Structures - Berkeley Math
  27. [PDF] Topology - I - UChicago Math
  28. [PDF] Zuoqin Wang Time: April 26, 2020 AXIOMS OF COUNTABILITY Last ...
  29. [PDF] “The Lindelöf Property” - University of Birmingham
  30. [PDF] Partitions of Unity
  31. S ey},L)y for U{S - American Mathematical Society
  32. https://math.uchicago.edu/~may/REU2023/REUPapers/Ho%2CAaron.pdf
  33. Smooth embeddings of the Long Line and other non-paracompact ...
  34. Examples of non-metrizable spaces - MathOverflow
  35. [PDF] Section 40. The Nagata-Smirnov Metrization Theorem
  36. [PDF] Nagata-Smirnov Metrization Theorem.nb - UChicago Math
  37. [PDF] 1 Motivation 2 Prerequisites
  38. [PDF] General Topology. Part 4: Metric Spaces - CSUSM
  39. [PDF] Hilbert spaces
  40. [PDF] Function Spaces 1 Hilbert Spaces
  41. [PDF] 3 | Open Sets
  42. [PDF] Review 1
  43. [PDF] Notes on Smooth Manifolds - IME-USP
  44. [PDF] Manifold Theory Peter Petersen - UCLA Mathematics
  45. [PDF] Stat 8112 Lecture Notes Weak Convergence in Metric Spaces
  46. [PDF] POLISH SPACES
  47. [PDF] A Brief Exploration of the Sorgenfrey Line - Auburn University
  48. [PDF] Metrizable Compactifications - Susan Klein
  49. [PDF] The Tychonoff Plank - G Eric Moorhouse
  50. [PDF] 13 | Metrization of Manifolds
  51. [PDF] Counterexamples in Topology - rexresearch1
  52. Fréchet space in nLab
  53. A fixed point theorem for Fréchet spaces - ScienceDirect.com
  54. [PDF] Functional Analysis, Sobolev Spaces and Partial Differential Equations
  55. [PDF] arXiv:1702.07867v1 [math.FA] 25 Feb 2017
  56. Semi-groups of operators in Fréchet space amd applications to partial differential equations
User Avatar
No comments yet.