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In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.[1][2]

Informal discussion

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For ordinary Euclidean spaces, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" dimension, and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by open sets.

In general, a topological space X can be covered by open sets, in that one can find a collection of open sets such that X lies inside of their union. The covering dimension is the smallest number n such that for every cover, there is a refinement in which every point in X lies in the intersection of no more than n + 1 covering sets. This is the gist of the formal definition below. The goal of the definition is to provide a number (an integer) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under homeomorphisms.

The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square.

Refinement of the cover of a circle
 The first image shows a refinement (on the bottom) of a colored cover (on the top) of a black circular line. Note how in the refinement, no point on the circle is contained in more than two sets, and also how the sets link to one another to form a "chain".
Refinement of the cover of a square
 The top half of the second image shows a cover (colored) of a planar shape (dark), where all of the shape's points are contained in anywhere from one to all four of the cover's sets. The bottom illustrates that any attempt to refine said cover such that no point would be contained in more than two sets—ultimately fails at the intersection of set borders. Thus, a planar shape is not "webby": it cannot be covered with "chains", per se. Instead, it proves to be thicker in some sense. More rigorously put, its topological dimension must be greater than 1.

Formal definition

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Henri Lebesgue used closed "bricks" to study covering dimension in 1921.[3]

The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.[4]

A modern definition is as follows. An open cover of a topological space X is a family of open sets Uα such that their union is the whole space, Uα = X. The order or ply of an open cover = {Uα} is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words Uα1 ∩ ⋅⋅⋅ ∩ Uαm+1 = for α1, ..., αm+1 distinct. A refinement of an open cover = {Uα} is another open cover = {Vβ}, such that each Vβ is contained in some Uα. The covering dimension of a topological space X is defined to be the minimum value of n such that every finite open cover of X has an open refinement with order n + 1. The refinement can always be chosen to be finite.[5] Thus, if n is finite, Vβ1 ∩ ⋅⋅⋅ ∩ Vβn+2 = for β1, ..., βn+2 distinct. If no such minimal n exists, the space is said to have infinite covering dimension.

As a special case, a non-empty topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets, meaning any point in the space is contained in exactly one open set of this refinement.

Examples

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The empty set has covering dimension −1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0.

Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps.

Similarly, any open cover of the unit disk in the two-dimensional plane can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two.

More generally, the n-dimensional Euclidean space has covering dimension n.

Properties

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  • Homeomorphic spaces have the same covering dimension. That is, the covering dimension is a topological invariant.
  • The covering dimension of a normal space X is if and only if for any closed subset A of X, if is continuous, then there is an extension of to . Here, is the n-dimensional sphere.
  • Ostrand's theorem on covering dimension. If X is a normal topological space and = {Uα} is a locally finite cover of X of order ≤ n + 1, then, for each 1 ≤ i ≤ n + 1, there exists a family of pairwise disjoint open sets i = {Vi,α} shrinking , i.e. Vi,α ⊆ Uα, and together covering X.[6]

Relationships to other notions of dimension

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  • For a paracompact space X, the covering dimension can be equivalently defined as the minimum value of n, such that every open cover of X (of any size) has an open refinement with order n + 1.[7] In particular, this holds for all metric spaces.
  • Lebesgue covering theorem. The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex.
  • The covering dimension of a normal space is less than or equal to the large inductive dimension.
  • The covering dimension of a paracompact Hausdorff space is greater or equal to its cohomological dimension (in the sense of sheaves),[8] that is, one has for every sheaf of abelian groups on and every larger than the covering dimension of .
  • In a metric space, one can strengthen the notion of the multiplicity of a cover: a cover has r-multiplicity n + 1 if every r-ball intersects with at most n + 1 sets in the cover. This idea leads to the definitions of the asymptotic dimension and Assouad–Nagata dimension of a space: a space with asymptotic dimension n is n-dimensional "at large scales", and a space with Assouad–Nagata dimension n is n-dimensional "at every scale".

See also

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Notes

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References

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Further reading

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Lebesgue covering dimension

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The Lebesgue covering dimension is a topological invariant that assigns a non-negative (or ) to a , representing its based on the complexity of its open covers. It is defined as the smallest nn such that every finite open cover of the space admits an open refinement where no point belongs to more than n+1n+1 sets in the refinement; if no such finite nn exists, the dimension is infinite. This captures the intuitive notion of by ensuring that covers can be refined without excessive overlap, generalizing classical Euclidean dimensions to abstract spaces. Originally introduced by in 1911 to study the non-embeddability of domains in Euclidean spaces of different dimensions, the concept used coverings by "bricks" (rectangular sets) to define dimension for compact subsets of Rk\mathbb{R}^k. Lebesgue's work established that the dimension of such sets equals the minimal nn where small-diameter closed covers have order at most n+1n+1, with the Lebesgue covering lemma guaranteeing a positive scale for uniform refinements in compact sets. In 1933, Eduard Čech extended this to arbitrary topological spaces via the modern covering dimension framework, making it applicable beyond metric settings. In separable metric spaces, the Lebesgue covering dimension coincides with the small and large inductive dimensions, providing a unified measure; for example, the Euclidean space Rn\mathbb{R}^n has dimension exactly nn. It satisfies additivity properties, such as dim⁡(X×Y)≤dim⁡X+dim⁡Y\dim(X \times Y) \leq \dim X + \dim Y for normal spaces, and plays a crucial role in embedding theorems: a compact metric space of dimension nn embeds into R2n+1\mathbb{R}^{2n+1}, but not always into lower dimensions. These features make it a foundational tool in dimension theory, influencing areas like algebraic topology and geometric analysis.

Introduction

Informal Description

The Lebesgue covering dimension offers an intuitive way to quantify the "size" or complexity of a by examining how it can be covered with open sets, focusing on the minimal level of overlap needed in such coverings. Imagine trying to cover a straight line with small intervals: it is always possible to arrange them so that no point lies in more than two intervals at once, reflecting the one-dimensional nature of the line. In contrast, covering a plane requires allowing overlaps of up to three sets at any point, akin to laying bricks in a wall where no single point is shared by more than three bricks, capturing the two-dimensional essence. This represents the smallest integer nn such that covers can be refined to avoid excessive overlaps beyond order n+1n+1, providing a topological measure that aligns with everyday geometric intuitions for familiar spaces. Henri Lebesgue introduced this concept in the early 20th century as part of his foundational work in measure theory, where he sought a robust notion of to extend the ideas of length, area, and volume to more abstract sets while ensuring consistency across transformations. Unlike geometric dimensions, which depend on specific coordinates or distances, the covering dimension is purely topological, remaining unchanged under homeomorphisms—continuous bijections with continuous inverses—that distort shapes without tearing or gluing. This invariance makes it particularly valuable for studying spaces where metric structures are absent or irrelevant, emphasizing intrinsic properties over embedded realizations. In essence, the Lebesgue covering dimension bridges intuitive geometry with abstract topology, foreshadowing the formal definition in terms of open cover refinements, and has become a cornerstone for understanding spatial complexity without relying on numerical measurements.

Historical Context

The concept of Lebesgue covering dimension traces its origins to Henri Lebesgue's contributions in 1911, where he introduced covering-based arguments to analyze the dimensionality of sets in the context of measure theory and integration. Lebesgue, building on his earlier work in developing the Lebesgue integral, used these coverings to distinguish the dimensions of Euclidean spaces, demonstrating that the unit cube in Rn\mathbb{R}^n could not be covered by fewer than n+1n+1 sets of sufficiently small diameter without overlaps exceeding a certain order. This approach provided an intuitive topological characterization of dimension tied to the geometric properties essential for integration over irregular sets. The formalization of the covering dimension within began with Urysohn's work starting in 1922, who extended Lebesgue's ideas to compact metric spaces by defining a related inductive invariant that aligned with covering properties. Urysohn's seminal papers "Sur les multiplicités cantoriennes" (Parts I and II), published posthumously in Fundamenta Mathematicae in 1925 and 1926, established foundational theorems on , including its additivity and behavior under mappings, laying the groundwork for a rigorous topological independent of metric structures. Independently, advanced this in 1926 through his paper "Allgemeine Räume und Cartesische Räume. I" in the Proceedings of the Royal Academy of , where he refined the covering approach by introducing the large inductive and proving its equivalence to Urysohn's small inductive in separable metric spaces, thus solidifying the covering as a key topological invariant. Post-1930s developments saw significant refinements, including Eduard Čech's 1933 formalization of the covering for arbitrary topological spaces. This culminated in and Henry Wallman's 1941 monograph Dimension Theory, which systematically extended the Lebesgue covering from metric spaces to arbitrary topological spaces and compiled proofs of its core properties. This work addressed limitations in earlier definitions and incorporated results from the , including the recognition of the 's homotopy invariance—meaning that continuous deformations preserve the covering —established through contributions by figures like Čech and others who linked it to cohomological characterizations. and Wallman's text became a , influencing subsequent by emphasizing the 's role in theorems and product spaces.

Definition

Core Definition

The Lebesgue covering dimension of a XX, denoted dim⁡X\dim X, is the smallest n≥−1n \geq -1 such that every finite open cover of XX admits an open refinement of order at most nn, where dim⁡X=−1\dim X = -1 if XX is empty and dim⁡X=∞\dim X = \infty if no such finite nn exists. This definition captures the intuitive notion of through the minimal overlap required in refinements of open covers, providing a topological invariant independent of any metric structure. The order of an open cover U\mathcal{U} of XX, denoted ord⁡U\operatorname{ord} \mathcal{U}, is the largest kk such that there exist k+1k+1 distinct sets in U\mathcal{U} with nonempty intersection, or equivalently, ord⁡U=sup⁡x∈X(∣{U∈U∣x∈U}∣−1)\operatorname{ord} \mathcal{U} = \sup_{x \in X} (|\{U \in \mathcal{U} \mid x \in U\}| - 1); if no finite kk bounds this, then ord⁡U=∞\operatorname{ord} \mathcal{U} = \infty. For instance, a cover has order 0 if its sets are pairwise disjoint (no point lies in more than one set), and order at most nn means no point lies in more than n+1n+1 sets. An open cover V\mathcal{V} refines U\mathcal{U} if every V∈VV \in \mathcal{V} is contained in some U∈UU \in \mathcal{U}. Formally, dim⁡X=inf⁡{n∈N0∪{∞}∣every finite open cover of X has an open refinement of order ≤n},\dim X = \inf \{ n \in \mathbb{N}_0 \cup \{\infty\} \mid \text{every finite open cover of } X \text{ has an open refinement of order } \leq n \}, where N0={0,1,2,… }\mathbb{N}_0 = \{0, 1, 2, \dots \}. For normal spaces, the condition applies specifically to finite open covers having open refinements of order less than or equal to nn, ensuring the definition aligns with broader topological properties. The construction proceeds inductively: dim⁡X≤0\dim X \leq 0 if every finite open cover has a disjoint refinement; assuming dim⁡X≤n−1\dim X \leq n-1, dim⁡X≤n\dim X \leq n if every finite open cover has a refinement where no point lies in more than n+1n+1 sets, building successive levels of controlled overlap.

Equivalent Characterizations

The Lebesgue covering dimension of a XX, denoted dim⁡X\dim X, admits several equivalent characterizations that facilitate its computation and theoretical analysis, particularly in the context of separable metric spaces. One fundamental equivalence is to the small inductive dimension, defined recursively as follows: \indX=−1\ind X = -1 if XX is empty; otherwise, \indX≤n\ind X \leq n if for every point x∈Xx \in X and every open neighborhood VV of xx, there exists an open neighborhood UU such that x∈U⊆Vx \in U \subseteq V and the boundary \FrU\Fr U satisfies \ind(\FrU)≤n−1\ind(\Fr U) \leq n-1; then \indX=n\ind X = n if \indX≤n\ind X \leq n but not \indX≤n−1\ind X \leq n-1, and \indX=∞\ind X = \infty otherwise. This holds for regular spaces, and in separable metric spaces, \indX=dim⁡X\ind X = \dim X. A related characterization involves the large inductive dimension, denoted \IndX\Ind X, which is defined for normal spaces: \IndX≤n\Ind X \leq n if for every closed subset A⊆XA \subseteq X and every open set V⊇AV \supseteq A, there exists an open set UU such that A⊆U⊆VA \subseteq U \subseteq V and \Ind(\FrU)≤n−1\Ind(\Fr U) \leq n-1. In separable metric spaces, the small and large inductive dimensions coincide, \indX=\IndX\ind X = \Ind X, and both equal the covering dimension dim⁡X\dim X. More generally, for metrizable spaces, the Katětov-Morita theorem establishes that \IndX=dim⁡X\Ind X = \dim X. Another equivalent formulation concerns embeddings into : a separable XX with dim⁡X≤n\dim X \leq n can be topologically embedded into R2n+1\mathbb{R}^{2n+1}. For compact s, this embedding can be realized as a closed , providing a geometric realization of the bound. This result underscores the topological invariance of the covering and its compatibility with finite-dimensional . Finally, the covering dimension can be characterized through approximations by simplicial complexes via nerves of open covers. Specifically, dim⁡X≤n\dim X \leq n if every finite open cover U\mathcal{U} of XX admits an open refinement V\mathcal{V} such that the nerve N(V)N(\mathcal{V})—the abstract simplicial complex with vertices corresponding to elements of V\mathcal{V} and simplices for finite intersections—is of dimension at most nn. Equivalently, for compact metric spaces, dim⁡X≤n\dim X \leq n if for every ε>0\varepsilon > 0, there exists a finite open cover with mesh less than ε\varepsilon whose nerve has dimension at most nn, or if XX admits an ε\varepsilon-mapping onto an nn-dimensional polyhedron for sufficiently small ε\varepsilon. These nerve-based criteria enable dimensional analysis through combinatorial and geometric approximations.

Examples and Applications

Euclidean Spaces

The Lebesgue covering dimension of the Euclidean space Rn\mathbb{R}^n is nn. This result follows from the fundamental theorem of dimension theory, which equates the small inductive dimension, large inductive dimension, and covering dimension for separable metric spaces like Rn\mathbb{R}^n, and establishes that all three are equal to nn. To verify that the dimension is at least nn, note that Rn\mathbb{R}^n contains an open set homeomorphic to the open nn-ball, whose covering dimension is nn as established for the nn-simplex using Sperner's lemma on triangulations to show the existence of a refinement with order at least n+1n+1. That the dimension is at most nn is shown by constructing explicit refinements of arbitrary open covers using partitions of unity and the Brouwer fixed-point theorem, ensuring that every open cover admits a refinement of order at most n+1n+1. For product spaces, the Lebesgue covering dimension satisfies dim⁡(Rm×Rk)=m+k\dim(\mathbb{R}^m \times \mathbb{R}^k) = m + k. This additivity holds because Rm×Rk\mathbb{R}^m \times \mathbb{R}^k is homeomorphic to Rm+k\mathbb{R}^{m+k}, whose is m+km+k by the result for Euclidean spaces. More directly, the product theorem for covering dimension states that for separable metric spaces XX and YY, dim⁡(X×Y)≤dim⁡X+dim⁡Y+1\dim(X \times Y) \leq \dim X + \dim Y + 1, but equality is achieved in this case via explicit refinement constructions: given an open cover U\mathcal{U} of Rm×Rk\mathbb{R}^m \times \mathbb{R}^k, one forms a refinement by taking products of open refinements of the projections onto Rm\mathbb{R}^m and Rk\mathbb{R}^k with orders at most m+1m+1 and k+1k+1, respectively, and using a to shrink sets so that their intersections have controlled multiplicity not exceeding m+k+1m+k+1. Subspaces of Euclidean spaces illustrate how the Lebesgue covering captures intrinsic rather than embedding. For the nn- Sn={x∈Rn+1∣∥x∥=1}S^n = \{x \in \mathbb{R}^{n+1} \mid \|x\| = 1\}, the is nn, despite being embedded in Rn+1\mathbb{R}^{n+1} of n+1n+1. This follows from the subspace theorem, which implies dim⁡(Sn)≤n\dim(S^n) \leq n as SnS^n is a compact nn-dimensional manifold, and the lower bound is established by noting that SnS^n contains open sets homeomorphic to Rn\mathbb{R}^n, requiring covers of order at least n+1n+1 in refinements. The contrast highlights that covering is invariant under homeomorphisms and independent of the ambient space, unlike the embedding .

Topological Manifolds

A of dimension nn is a that is locally homeomorphic to Rn\mathbb{R}^n, and under standard assumptions such as second countability, its Lebesgue covering is exactly nn. This follows from the fact that every point has a neighborhood homeomorphic to an open subset of Rn\mathbb{R}^n, which has covering nn, and the use of partitions of unity allows for the construction of refinements of open covers that satisfy the condition globally. More precisely, for metrizable manifolds, the Lebesgue covering coincides with the small inductive , confirming that dim⁡(M)=n\dim(M) = n for an nn-manifold MM. Representative examples illustrate this invariance. The 2-torus T2T^2, obtained as the quotient R2/Z2\mathbb{R}^2 / \mathbb{Z}^2, is a compact 2-dimensional manifold and thus has Lebesgue covering dimension 2. Similarly, the Klein bottle, a non-orientable compact surface homeomorphic to the quotient of the square with opposite sides identified in a twisted manner, also has dimension 2, as its atlas consists of charts to open sets in R2\mathbb{R}^2. Pathological cases highlight spaces that achieve dimension 1 in the covering sense without satisfying the local Euclidean condition of manifolds. The long line, constructed as the on [ω1)×[0,1)[\omega_1) \times [0,1) with the order topology (excluding the endpoint), is a connected 1-dimensional Hausdorff space that is locally homeomorphic to R\mathbb{R} but fails second countability and paracompactness, yet its Lebesgue covering dimension is 1 due to its linear structure allowing order-based cover refinements. The pseudo-arc, a hereditarily indecomposable chainable continuum in the plane, is another example of covering dimension 1, as every open cover refines to one where no three sets intersect, but it is nowhere locally Euclidean, exhibiting wild embedding properties. The Lebesgue covering dimension plays a key role in the , as it is a topological invariant preserved under , enabling the distinction of manifolds up to homeomorphism based on local and global covering properties. In embedding theorems, it provides the bound for realization in : by the Menger–Nöbeling theorem, any compact of covering dimension nn, including compact nn-manifolds, embeds in R2n+1\mathbb{R}^{2n+1}, with implications for where smooth structures are imposed atop the topological framework.

Properties

Fundamental Properties

The Lebesgue covering dimension, denoted dim⁡X\dim X for a XX, is a topological invariant. Specifically, if f:X→Yf: X \to Y is a , then dim⁡X=dim⁡Y\dim X = \dim Y. This follows directly from the , as homeomorphisms preserve the refinement properties of open covers. In normal spaces, this invariance is established in Theorem 1.6.7. The also exhibits monotonicity with respect to subspaces. For any subspace A⊆XA \subseteq X, dim⁡A≤dim⁡X\dim A \leq \dim X. This property holds in regular spaces for the inductive dimension (Theorem 1.1.2) and extends to the covering dimension in normal and separable metric spaces (Theorems 1.7.7 and 4.1.7). Additionally, for the of two spaces, dim⁡(X⊔Y)=max⁡(dim⁡X,dim⁡Y)\dim(X \sqcup Y) = \max(\dim X, \dim Y). This additivity for disjoint unions is a consequence of the covering refinement condition applying separately to each component (Theorem 1.5.2 for disjoint subspaces). In separable metric spaces with dim⁡X≥1\dim X \geq 1, the dimension is stable under removal of a single point: dim⁡(X∖{p})=dim⁡X\dim(X \setminus \{p\}) = \dim X for any p∈Xp \in X. This result, given in Corollary 1.5.6, relies on the ability to refine covers while avoiding the point without increasing the order. For compact Hausdorff spaces, the Lebesgue covering coincides with the small inductive dimension, providing a unified finite or infinite value that characterizes the space's topological complexity (Theorem 1.9.9). Moreover, if dim⁡X≤n\dim X \leq n, then XX admits an open cover that refines to one of order at most n+1n+1, aiding paracompact decompositions in normal spaces with finite dimension.

Metric Space Extensions

In metric spaces, the existence of Lebesgue numbers for open covers of compact subsets facilitates the construction of uniform refinements that preserve the covering dimension. Specifically, for a compact metric space XX, every open cover U\mathcal{U} admits a Lebesgue number δ>0\delta > 0 such that any subset of diameter less than δ\delta is contained in some member of U\mathcal{U}; this property ensures that refinements can be chosen with uniformly small mesh (maximum diameter) while maintaining order at most n+1n+1 if dim⁡X≤n\dim X \leq n, thereby extending the topological invariance of dimension to metric-specific uniform structures. A key extension concerns product spaces. In general topological spaces, the Lebesgue covering dimension satisfies dim⁡(X×Y)≤dim⁡X+dim⁡Y+1\dim(X \times Y) \leq \dim X + \dim Y + 1. For compact metric spaces, this inequality holds, and while additivity dim⁡(X×Y)=dim⁡X+dim⁡Y\dim(X \times Y) = \dim X + \dim Y occurs in many cases (e.g., for Euclidean spaces or manifolds), counterexamples exist where the product dimension is strictly less, though the +1+1 bound remains sharp overall. The dimension provides another metric refinement, defined as the smallest nn such that there exists a light map f:X→Rnf: X \to \mathbb{R}^n, where "light" means that preimages of small-diameter sets have controlled diameter proportional to the original. For any XX, the dimension satisfies dim⁡Lip(X)≥dim⁡X\dim_\text{Lip}(X) \geq \dim X, with equality holding for doubling spaces, where the metric structure allows bi- equivalence to spaces of matching topological . In applications to fractals, the Assouad —measuring the worst-case scaling via covers of balls—yields upper bounds for the Lebesgue covering , as dim⁡X≤dim⁡AX\dim X \leq \dim_A X for metric spaces XX. This relation is particularly useful for non-integer-dimensional fractals like self-similar sets, where dim⁡X=0\dim X = 0 but dim⁡AX>0\dim_A X > 0, providing bounds without equality in overlapping cases; for instance, in self-similar fractals violating the weak separation property, dim⁡AF=1\dim_A F = 1 while the covering remains 0.

Comparisons

Inductive Dimension

The small inductive dimension, denoted ind(X), of a topological space X is defined recursively as follows: ind(∅) = -1, and ind(X) ≤ 0 if for every point x ∈ X and every open neighborhood U of x, there exists an open neighborhood V of x with V ⊆ U such that the boundary ∂V = cl(V) \ V is empty. For n ≥ 1, ind(X) ≤ n if for every point x ∈ X and every open neighborhood U of x, there exists an open neighborhood V of x with V ⊆ U such that ind(∂V) ≤ n-1. The value ind(X) is the smallest nonnegative integer n satisfying ind(X) ≤ n, or ∞ if no such n exists. The large inductive dimension, denoted Ind(X), is defined similarly but applies to closed sets: Ind(∅) = -1, and Ind(X) ≤ 0 if for every closed set F ⊆ X and every open set U ⊇ F, there exists an open set V with F ⊆ V ⊆ U such that ∂V is empty. For n ≥ 1, Ind(X) ≤ n if for every closed set F ⊆ X and every open set U ⊇ F, there exists an open set V with F ⊆ V ⊆ U such that Ind(∂V) ≤ n-1. The value Ind(X) is the smallest nonnegative integer n satisfying Ind(X) ≤ n, or ∞ if no such n exists. In normal spaces, Ind(X) coincides with the Lebesgue covering dimension dim(X). For any X, the inequalities ind(X) ≤ dim(X) ≤ Ind(X) hold. A brief proof sketch proceeds as follows: the inequality ind(X) ≤ dim(X) follows from the fact that the small inductive satisfies the conditions for order of open covers, allowing a refinement argument similar to the covering definition; conversely, dim(X) ≤ Ind(X) arises because the large inductive 's boundary condition implies the existence of refinements with controlled order via normality, ensuring no overlaps exceeding the bound. These three dimensions agree in separable metric spaces, where ind(X) = dim(X) = Ind(X). The small and large inductive dimensions may differ in certain pathological compacta. For instance, V. V. Filippov constructed a compact with ind(X) = 0 but Ind(X) = 1, resolving a problem of P. S. Aleksandrov by showing that the inequality ind(X) ≤ Ind(X) can be strict.

Other Dimension Theories

The Lebesgue covering , also known as the topological , provides a lower bound for the in separable s. Specifically, for any separable XX, dim⁡X≤dim⁡HX\dim X \leq \dim_H X, where dim⁡\dim denotes the covering and dim⁡H\dim_H the . This inequality arises because the covering captures the minimal number of coordinates needed to separate points via open covers, while the accounts for metric scaling properties that can yield non- values greater than or equal to the topological one. Equality holds in cases like Euclidean spaces or smooth manifolds, where both dimensions equal the classical . For self-similar sets satisfying the open set condition, the often equals the similarity dimension, though typically the Hausdorff dimension exceeds the topological one for fractals, as in the Sierpinski gasket where the topological dimension is 1 but the is log⁡23≈1.585\log_2 3 \approx 1.585. However, strict inequality occurs in examples like the Cantor dust, the product of two middle-thirds Cantor sets in the plane, which has topological dimension 0 (being totally disconnected) but 2log⁡2/log⁡3≈1.26192 \log 2 / \log 3 \approx 1.2619. In the context of fractal dimensions, the Lebesgue covering dimension serves as a lower bound for the box-counting dimension (also known as the Minkowski-Bouligand dimension) in metric spaces. The box-counting dimension, defined as lim⁡δ→0log⁡N(δ)−log⁡δ\lim_{\delta \to 0} \frac{\log N(\delta)}{-\log \delta} where N(δ)N(\delta) is the minimal number of sets of diameter δ\delta covering the space, satisfies dim⁡X≤dim⁡BX\dim X \leq \dim_B X, reflecting how topological separation requires at least as many boxes as the covering order suggests. This bound is sharp for spaces like the Cantor set, where dim⁡=0\dim = 0 but dim⁡B=log⁡2/log⁡3≈0.631\dim_B = \log 2 / \log 3 \approx 0.631. Embedding dimensions, such as the minimal Euclidean dimension for isometric embedding, are similarly bounded below by the topological dimension, ensuring that fractals with low covering dimension can still require higher embedding spaces due to metric distortions. The Lebesgue covering dimension relates closely to the cohomological dimension in , particularly for paracompact spaces. For a paracompact space XX, dim⁡X≤n\dim X \leq n if and only if the Čech cohomology groups Hn+1(Y;G)=0H^{n+1}(Y; G) = 0 for every closed subset Y⊆XY \subseteq X and every coefficient group GG. This equivalence, established via sheaf cohomology vanishing, allows topological dimension to inform homological computations in paracompacta like manifolds. In CW-complexes, a key structure in algebraic topology, the Lebesgue covering dimension coincides with the cellular dimension, defined as the highest index nn such that the nn-skeleton is non-empty. For a CW-complex XX, dim⁡X=n\dim X = n if and only if XX is homotopy equivalent to its nn-skeleton, with no higher cells needed. This alignment facilitates homotopy theory applications, where the dimension bounds the range of non-trivial homotopy groups; aiding spectral sequence computations and embedding theorems.

References

  1. https://arxiv.org/pdf/2110.08867
  2. https://personal.colby.edu/~sataylor/teaching/F14/MA331/TopologicalDimension.pdf
  3. https://link.springer.com/article/10.1007/BF01461155
  4. https://www.jstor.org/stable/3621569
  5. https://pi.math.cornell.edu/~steinhurst/docs/dimension.pdf
  6. https://mathworld.wolfram.com/LebesgueCoveringDimension.html
  7. https://www.britannica.com/biography/Henri-Leon-Lebesgue
  8. https://projecteuclid.org/download/pdf_1/euclid.bams/1183493602
  9. https://mathshistory.st-andrews.ac.uk/Biographies/Urysohn/
  10. https://link.springer.com/content/pdf/10.1007/978-3-7091-6110-4.pdf
  11. https://press.princeton.edu/books/hardcover/9780691653686/dimension-theory
  12. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf
  13. https://mathoverflow.net/questions/228574/two-definitions-of-lebesgue-covering-dimension
  14. https://www.maths.gla.ac.uk/~mpowell/topological-manifolds-lecture-notes.pdf
  15. https://www.mscsnet.mu.edu/~paul/Paper/tamapprop042916.pdf
  16. https://arxiv.org/pdf/2305.17717
  17. https://arxiv.org/pdf/1002.0549
  18. https://www.sciencedirect.com/science/article/pii/S0019357717300472
  19. https://www.pnas.org/doi/pdf/10.1073/pnas.75.10.4671
  20. https://arxiv.org/pdf/1908.04421
  21. https://par.nsf.gov/servlets/purl/10214997
  22. https://arxiv.org/pdf/2005.03763
  23. https://iopscience.iop.org/article/10.1070/SM1970v012n01ABEH000909
  24. https://topology.nipissingu.ca/tp/reprints/v11/tp11209s.pdf
  25. https://jameshbailie.github.io/Papers/papers/Hausdorff.pdf
  26. https://u.math.biu.ac.il/~megereli/final_topology.pdf
  27. https://epub.ub.uni-muenchen.de/4524/1/4524.pdf
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