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In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.[1] A graphical illustration of a zero-dimensional space is a point.[2]

Definition

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Specifically:

  • A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement that is a cover by disjoint open sets.
  • A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.
  • A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.

The three notions above agree for separable, metrisable spaces (see Inductive dimension § Relationships between dimensions).

Properties of spaces with small inductive dimension zero

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Manifolds

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All points of a zero-dimensional manifold are isolated.

Notes

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References

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Zero-dimensional space

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In , a zero-dimensional space is a fundamental concept denoting a space with dimension zero, lacking extent in any direction. In , this is represented by a point, an undefined primitive that indicates location without length, width, height, or any measurable size. In , a zero-dimensional space is defined as a possessing a basis of clopen sets—subsets that are both open and closed—allowing for a highly disconnected structure where points can be separated by such sets. Zero-dimensional spaces exhibit key properties that underscore their minimal connectivity. Every zero-dimensional space is totally disconnected, meaning the only connected subsets are singletons, preventing any nontrivial continuous paths between distinct points. In Hausdorff zero-dimensional spaces, this structure implies complete regularity, enabling the space to be embedded into a product of intervals via continuous functions. Metric zero-dimensional spaces include the compact or the non-compact irrationals endowed with the from the reals, and serve as canonical examples, demonstrating how infinite yet "point-like" in dimensionality these spaces can be. These spaces play a crucial role across mathematical disciplines. In geometry, the point as a zero-dimensional object forms the foundational building block for higher-dimensional constructs like lines (one-dimensional) and planes (two-dimensional). In algebraic contexts, zero-dimensional ideals in polynomial rings correspond to finite varieties, reducing systems of equations to isolated solutions. Topologically, discrete spaces on finite or countable sets are zero-dimensional, and the concept extends to profinite spaces in scheme theory, where affine opens have discrete topologies. Overall, zero-dimensional spaces highlight the boundary of spatial intuition, bridging pure abstraction with applications in analysis, , and beyond.

Definition and Characterizations

Formal Topological Definition

In , a zero-dimensional topological space XX is formally defined as one that admits a basis consisting entirely of clopen sets. This basis ensures that every point in XX has arbitrarily small neighborhoods that are both open and closed relative to the space, providing a foundational characterization of dimension zero without reference to metric or inductive structures. A in a XX is a that is simultaneously open and closed; the ∅\emptyset and the entire space XX serve as the trivial examples of clopen sets in any . This definition presupposes familiarity with the standard concepts of topological spaces, where open and closed sets are defined via the , and bases, which are collections of open sets such that every is a union of basis elements. The concept of zero-dimensionality via a clopen basis was first formalized in the early 1920s within the broader framework of dimension theory, independently developed by Pavel Urysohn and as part of their efforts to axiomatize topological dimension for general spaces.

Equivalent Characterizations

A XX is zero-dimensional if and only if every point x∈Xx \in X has a local basis consisting of clopen neighborhoods. To see this equivalence with the existence of a global clopen basis, note that if XX has a clopen basis, then for any open neighborhood UU of xx, there exists a basis element B⊆UB \subseteq U that is clopen and contains xx, forming the local basis at xx. Conversely, if every point has such a local clopen basis, the union of all these local clopen neighborhoods over all points in XX constitutes a clopen basis for the entire , as any open set UU can be covered by such neighborhoods contained within it. In Hausdorff spaces, zero-dimensionality is equivalent to the space being totally disconnected and having a clopen basis. A space is totally disconnected if the only connected subsets are singletons, meaning its quasi-components are points. The key step in one direction relies on the T1 : for distinct points x,y∈Xx, y \in X, there exists an open neighborhood UU of xx not containing yy, and a clopen basis element B⊆UB \subseteq U containing xx but not yy, separating them clopenly and implying total disconnectedness. The converse holds by definition, as the clopen basis provides the additional structure beyond mere disconnectedness, which alone does not suffice, as there exist totally disconnected Hausdorff spaces without a clopen basis (e.g., Sierpiński's example). This topological characterization via clopen bases extends to non-Hausdorff spaces, where zero-dimensionality is defined without requiring T1 separation, but the equivalences to local clopen bases and total disconnectedness may fail without additional axioms like T1, as clopen sets might not adequately separate non-closed points.

Properties in Dimension Theory

Small Inductive Dimension

The small inductive dimension of a XX, denoted ind⁡(X)\operatorname{ind}(X), is defined recursively as follows: ind⁡(X)=−1\operatorname{ind}(X) = -1 if XX is empty; for n≥0n \geq 0, ind⁡(X)≤n\operatorname{ind}(X) \leq n if every point x∈Xx \in X has arbitrarily small neighborhoods VV containing an open set UU with x∈U⊆Vx \in U \subseteq V such that the boundary Fr⁡U\operatorname{Fr} U satisfies ind⁡(Fr⁡U)≤n−1\operatorname{ind}(\operatorname{Fr} U) \leq n-1; and ind⁡(X)=n\operatorname{ind}(X) = n if ind⁡(X)≤n\operatorname{ind}(X) \leq n but not ind⁡(X)≤n−1\operatorname{ind}(X) \leq n-1, with ind⁡(X)=∞\operatorname{ind}(X) = \infty if it exceeds all finite nn. For zero-dimensional spaces, ind⁡(X)=0\operatorname{ind}(X) = 0 (assuming XX is non-empty) means that for every point and neighborhood, there exist arbitrarily small open sets with empty boundaries, i.e., clopen sets. In regular topological spaces, ind⁡(X)=0\operatorname{ind}(X) = 0 if and only if XX has a basis consisting of clopen sets; conversely, the existence of such a clopen basis implies ind⁡(X)=0\operatorname{ind}(X) = 0. This characterization highlights the small inductive dimension as a boundary-based invariant that captures the absence of higher-dimensional "layers" around points. Spaces with ind⁡(X)=0\operatorname{ind}(X) = 0 are hereditarily disconnected, meaning no subspace contains a connected component with more than one point, as any potential connected subset would require boundaries of positive in the inductive construction. This property underscores their role as the atomic base case in the inductive dimension hierarchy, where higher dimensions are built by iteratively adjoining boundaries of successively lower-dimensional spaces, starting from the empty or point-like zero level.

Covering and Large Inductive Dimensions

The covering dimension of a XX, denoted dim⁡X\dim X, is defined inductively. Specifically, dim⁡X≤0\dim X \leq 0 every open cover of XX admits an open refinement consisting of pairwise disjoint sets. This condition implies that XX can be "separated" into disjoint open components without overlap in any refinement, capturing a minimal topological complexity. In normal spaces, the covering dimension dim⁡X=0\dim X = 0 if and only if XX has a basis consisting of clopen sets. This equivalence highlights the role of normality in ensuring that refinements can be chosen to align with the clopen structure, providing a concrete characterization of zero-dimensionality beyond the abstract covering condition. The large inductive dimension, denoted Ind⁡X\operatorname{Ind} X, is defined for normal spaces using boundaries of separating sets. Specifically, Ind⁡X≤0\operatorname{Ind} X \leq 0 (and thus Ind⁡X=0\operatorname{Ind} X = 0 for non-empty XX) if, for every closed subset A⊂XA \subset X and every open set V⊃AV \supset A, there exists an open set UU such that A⊂U⊂VA \subset U \subset V and cl⁡U∖U=∅\operatorname{cl} U \setminus U = \emptyset, meaning UU is clopen. This boundary condition differs from the small inductive dimension by applying to arbitrary closed-open pairs rather than point neighborhoods, but yields a similar outcome for zero-dimensionality: Ind⁡X=0\operatorname{Ind} X = 0 if and only if XX has a clopen basis. In compact Hausdorff spaces, the large inductive dimension Ind⁡X\operatorname{Ind} X coincides with the small inductive dimension ind⁡X\operatorname{ind} X. More broadly, all standard dimension theories agree on zero-dimensionality in metrizable spaces: for separable metric spaces, dim⁡X=ind⁡X=Ind⁡X=0\dim X = \operatorname{ind} X = \operatorname{Ind} X = 0. This unification ensures that the covering and inductive approaches yield identical results for zero-dimensional metrizable spaces, such as the rationals or .

Topological and Geometric Properties

Separation and Connectedness

A zero-dimensional , defined as a non-empty T1T_1- with a basis consisting of clopen sets, is hereditarily disconnected, meaning that no subspace contains a connected with more than one point. This property ensures that the only connected components of the are singletons, rendering the totally disconnected. Consequently, every subspace with at least two points is disconnected, as it can be expressed as the union of two disjoint non-empty open sets. Regarding separation axioms, zero-dimensional spaces satisfy the T1T_1 condition by definition and are typically T0T_0 in broader contexts, but they excel in providing clopen separations for distinct points. For any two distinct points xx and yy in such a space, there exists a UU containing xx but not yy, due to the clopen basis. This allows the space to be partitioned into clopen sets that separate any pair of points, analogous to the discrete separation in digital topologies. The hereditary nature of disconnection in zero-dimensional spaces underscores their structural simplicity, where connectedness fails at every scale beyond individual points. In Hausdorff zero-dimensional spaces, this total disconnection is particularly pronounced, aligning with stronger separation properties like those in Tychonoff spaces.

Basis and Compactness

A zero-dimensional topological space possesses a basis consisting of clopen sets, meaning that for every point and every neighborhood thereof, there exists a clopen neighborhood contained within it. This structural feature ensures that any basis for the can be refined to one composed entirely of clopen sets, as the clopen basis elements can be used to subdivide arbitrary open sets without introducing nested proper open subsets that lack clopen boundaries. Such refinement highlights the absence of dimensional depth, where separations occur immediately at the clopen level rather than requiring iterative refinements as in higher-dimensional spaces. In compact , zero-dimensionality is equivalent to being totally disconnected. Specifically, every countable compact Hausdorff space is homeomorphic to a countable successor ordinal under the or to a finite , as established by the Mazurkiewicz–Sierpiński theorem; this classification applies since such spaces are zero-dimensional. In non-countable settings, these spaces may include examples like the , which are not homeomorphic to ordinals. For metric zero-dimensional spaces, yields second countability, as every compact admits a countable basis, thereby ensuring the clopen basis is also countable and enhancing properties like separability and the Lindelöf condition. This second countability strengthens metrizability, allowing uniform control over the topology via countable clopen covers. Unlike higher-dimensional spaces, where products increase the overall , zero-dimensionality is preserved under arbitrary products: the product of zero-dimensional spaces inherits a clopen basis from the . Consequently, compactness in such products is maintained via , enabling infinite products of compact zero-dimensional spaces to remain compact and zero-dimensional.

Examples

Discrete and Finite Spaces

Finite sets equipped with the discrete topology provide the most straightforward examples of zero-dimensional spaces. In the discrete topology on a XX, every is open by , making singletons {x}\{x\} both open and closed (clopen). These singletons form a basis for the topology, satisfying the condition for small inductive dimension ind⁡X=0\operatorname{ind} X = 0.793-818.pdf) In such spaces, every is clopen, and there are no nontrivial connected subsets; the connected components are exactly the singletons, reflecting the total disconnectedness inherent to zero-dimensionality. The natural numbers N\mathbb{N} with the discrete topology illustrate an infinite zero-dimensional space. Here, singletons again form a clopen basis, ensuring ind⁡N=0\operatorname{ind} \mathbb{N} = 0, but the space lacks compactness, as no finite subcover exists for the open cover by singletons.793-818.pdf) Discrete spaces, whether finite or countably infinite, model point clouds in introductory topology, treating collections of isolated points without imposing additional geometric structure.

Pathological and Fractal Examples

One prominent pathological example of a zero-dimensional space is the , constructed by iteratively removing the middle third of the interval [0,1]. The resulting set CC is compact, metrizable, totally disconnected, and uncountable, yet it admits a basis consisting of clopen sets derived from the complements of the removed open intervals during its construction. This clopen basis confirms its zero-dimensionality, despite its -like structure and positive of log⁡2/log⁡3≈0.631\log 2 / \log 3 \approx 0.631, illustrating how topological can differ from measure-theoretic notions. The rational numbers Q\mathbb{Q}, endowed with the subspace topology inherited from the real line R\mathbb{R}, provide another counterintuitive zero-dimensional space. Although singletons are not open in Q\mathbb{Q} due to its dense embedding in R\mathbb{R}, the space possesses a basis of clopen sets formed by intersecting open intervals in R\mathbb{R} with Q\mathbb{Q}; each such intersection is both open and closed in Q\mathbb{Q} because its complement in Q\mathbb{Q} is a union of similar intersections with the removed irrational parts. This structure renders Q\mathbb{Q} zero-dimensional and countable without isolated points, highlighting its totally disconnected nature in a seemingly dense context. The irrational numbers P=R∖Q\mathbb{P} = \mathbb{R} \setminus \mathbb{Q}, also with the subspace topology from R\mathbb{R}, form a zero-dimensional space that is homeomorphic to the NN\mathbb{N}^\mathbb{N}, the set of all sequences of natural numbers equipped with the where N\mathbb{N} is discrete. The is zero-dimensional as a countable product of zero-dimensional discrete spaces, featuring a clopen basis of "cylinders" defined by finite initial segments of sequences, and it is completely metrizable and non-locally compact. This underscores the pathological density of irrationals, which lack isolated points yet maintain zero-dimensionality through total disconnection. The field of p-adic numbers Qp\mathbb{Q}_p, for a prime p, equipped with its natural ultrametric topology, exemplifies a zero-dimensional outside the archimedean setting. It is totally disconnected and locally compact, with a basis of clopen balls given by {x∈Qp:∣x−a∣p<p−n}\{ x \in \mathbb{Q}_p : |x - a|_p < p^{-n} \} for a∈Qpa \in \mathbb{Q}_p and n∈Nn \in \mathbb{N}, where ∣⋅∣p|\cdot|_p is the . The p-adic integers Zp\mathbb{Z}_p, a compact open of Qp\mathbb{Q}_p, further illustrate this by being homeomorphic to the , reinforcing the zero-dimensional character through its profinite structure.

Manifolds and Applications

Zero-Dimensional Manifolds

A zero-dimensional manifold is defined as a in which every point has a neighborhood homeomorphic to R0\mathbb{R}^0, the zero-dimensional consisting of a single point. This condition implies that each point is isolated, making the space discrete. Topological manifolds of dimension zero are second-countable, Hausdorff spaces that are locally Euclidean of dimension zero. The second-countability ensures that the space has a countable basis, which, combined with the discrete topology, restricts such manifolds to countable discrete sets. Finite sets of points, such as the zero S0={−1,1}⊂RS^0 = \{-1, 1\} \subset \mathbb{R}, serve as representative examples of these structures. The smooth structure on a zero-dimensional manifold is trivial, as charts map points to the sole element of R0\mathbb{R}^0, with transition maps being the identity on a point. Consequently, the at each point is the zero-dimensional {0}\{0\}, rendering differentiable concepts inapplicable in a non-trivial sense. These manifolds are classified as either finite or countably infinite discrete spaces, distinguishing them from uncountable discrete sets that fail second countability.

Role in Geometry and Analysis

In , zero-dimensional submanifolds appear as discrete collections of isolated points embedded in higher-dimensional spaces. For a smooth f:M→Nf: M \to N between manifolds, the preimage f−1(y)f^{-1}(y) of a regular value y∈Ny \in N forms a zero-dimensional , consisting of a of points when MM is compact. These point sets are essential in the study of submanifolds, where they represent the lowest-dimensional components in decompositions of geometric objects. In singularity theory, zero-dimensional spaces arise as the lowest strata in stratifications of singular varieties or mappings, where the local drops to isolated points at points of highest degeneracy. Such strata capture the most severe singularities, such as nodes or cusps reduced to point-like loci in resolution processes. This dimensional drop facilitates the classification of singularities by their stratified structure, enabling analysis of stability and bifurcations in geometric configurations. In , zero-dimensional measures like the Dirac delta measure δx\delta_x at a point xx model concentrated masses or impulses supported on zero-dimensional sets. The Dirac delta satisfies ⟨δx,f⟩=f(x)\langle \delta_x, f \rangle = f(x) for smooth test functions ff, and its support {x}\{x\} is intrinsically zero-dimensional. More broadly, distributions with zero-dimensional support consist of finite sums of Dirac deltas and their at those points, providing tools for solving partial differential equations with point sources, such as in or wave propagation. These objects extend classical measures to handle singularities where mass collapses to points. In , the Spec⁡(A)\operatorname{Spec}(A) of an AA defines a zero-dimensional scheme, which decomposes into a finite of points, each equipped with structure from a local Artinian of finite dimension over the base field. For a zero-dimensional algebraic scheme over a field kk, it is affine and finite over Spec⁡(k)\operatorname{Spec}(k), modeling finite point configurations with multiplicities in . Zero-dimensional spaces also model "dust"-like structures in dynamical systems and fractals, representing totally disconnected invariant sets in chaotic dynamics. In , subshifts on zero-dimensional spaces capture the combinatorial complexity of iterations, as surveyed in techniques for handling such systems via array representations and inverse limits. These models underpin and the study of attractors with zero topological dimension but positive in geometry.

References

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