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Coarse structure

Concept in geometry and topology

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Source updated May 15, 2026

In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structure of metric spaces and topological spaces to be defined.

The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space, such as boundedness, or the degrees of freedom of the space, do not depend on such features. Coarse geometry and coarse topology provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.

Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.

01Definition

A coarse structure on a set $X$ is a collection $\mathbf {E}$ of subsets of $X\times X$ (therefore falling under the more general categorization of binary relations on $X$ ) called controlled sets, and so that $\mathbf {E}$ possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:

Identity/diagonal:
The diagonal $\Delta =\{(x,x):x\in X\}$ is a member of $\mathbf {E}$ , the identity relation.
Closed under taking subsets:
If $E\in \mathbf {E}$ and $F\subseteq E,$ then $F\in \mathbf {E} .$
Closed under taking inverses:
If $E\in \mathbf {E}$ then the inverse (or transpose) $E^{-1}=\{(y,x):(x,y)\in E\}$ is a member of $\mathbf {E}$ , the inverse relation.
Closed under taking unions:
If $E,F\in \mathbf {E}$ then their union $E\cup F$ is a member of $\mathbf {E} .$
Closed under composition:
If $E,F\in \mathbf {E}$ then their product $E\circ F=\{(x,y):{\text{ there exists }}z\in X{\text{ such that }}(x,z)\in E{\text{ and }}(z,y)\in F\}$ is a member of $\mathbf {E}$ , the composition of relations.

A set $X$ endowed with a coarse structure $\mathbf {E}$ is a coarse space.

Let $E\in \mathbf {E}$ a controlled set. For a subset $K$ of $X,$ the set $E[K]$ is defined as $\{x\in X:(x,k)\in E{\text{ for some }}k\in K\}.$ We define the section of $E$ by $x$ to be the set $E[\{x\}],$ also denoted $E_{x}.$ The symbol $E^{y}$ denotes the set $E^{-1}[\{y\}].$ These are forms of projections.

A subset $B$ of $X$ is said to be a bounded set if $B\times B$ is a controlled set.

Intuition

The controlled sets are "small" sets, or "negligible sets": a set $A$ such that $A\times A$ is controlled is negligible, while a function $f:X\to X$ such that its graph is controlled is "close" to the identity. In the bounded coarse structure, these sets are the bounded sets, and the functions are the ones that are a finite distance from the identity in the uniform metric.

02Coarse maps

Given a set $S$ and a coarse structure $X,$ we say that the maps $f:S\to X$ and $g:S\to X$ are close if $\{(f(s),g(s)):s\in S\}$ is a controlled set.

For coarse structures $X$ and $Y,$ we say that $f:X\to Y$ is a coarse map if for each bounded set $B$ of $Y$ the set $f^{-1}(B)$ is bounded in $X$ and for each controlled set $E$ of $X$ the set $(f\times f)(E)$ is controlled in $Y.$ $X$ and $Y$ are said to be coarsely equivalent if there exists coarse maps $f:X\to Y$ and $g:Y\to X$ such that $f\circ g$ is close to $\operatorname {id} _{Y}$ and $g\circ f$ is close to $\operatorname {id} _{X}.$

03Examples

The bounded coarse structure on a metric space $(X,d)$ is the collection $\mathbf {E}$ of all subsets $E$ of $X\times X$ such that $\sup _{(x,y)\in E}d(x,y)$ is finite. With this structure, the integer lattice $\mathbb {Z} ^{n}$ is coarsely equivalent to $n$ -dimensional Euclidean space.
A space $X$ where $X\times X$ is controlled is called a bounded space. Such a space is coarsely equivalent to a point. A metric space with the bounded coarse structure is bounded (as a coarse space) if and only if it is bounded (as a metric space).
The trivial coarse structure only consists of the diagonal and its subsets. In this structure, a map is a coarse equivalence if and only if it is a bijection (of sets).
The $C_{0}$ coarse structure on a metric space $(X,d)$ is the collection of all subsets $E$ of $X\times X$ such that for all $\varepsilon >0$ there is a compact set $K$ of $E$ such that $d(x,y)<\varepsilon$ for all $(x,y)\in E\setminus K\times K.$ Alternatively, the collection of all subsets $E$ of $X\times X$ such that ${\overline {\{(x,y)\in E:d(x,y)\geq \varepsilon \}}}$ is compact.
The discrete coarse structure on a set $X$ consists of the diagonal $\Delta$ together with subsets $E$ of $X\times X$ which contain only a finite number of points $(x,y)$ off the diagonal.
If $X$ is a topological space then the indiscrete coarse structure on $X$ consists of all proper subsets of $X\times X,$ meaning all subsets $E$ such that $E[K]$ and $E^{-1}[K]$ are relatively compact whenever $K$ is relatively compact.

04Bounded sets

Let ${\mathcal {B}}=\{B\subseteq X:B\times B\in \mathbf {E} \}$ be the collection of all bounded sets of a coarse space $X$ . Say that a coarse structure $\mathbf {E}$ on $X$ is coarsely connected if $\{(x,y)\}\in \mathbf {E}$ for all $(x,y)\in X\times X$ . Then ${\mathcal {B}}$ is a bornology on $X$ if and only if $\mathbf {E}$ is coarsely connected. For example, if $X$ has at least two points and $\mathbf {E}$ is the trivial coarse structure, then ${\mathcal {B}}$ is not a bornology. Bounded, discrete and indiscrete coarse structures are coarsely connected.

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Sources and credits

This article is adapted from the Wikipedia article “Coarse structure”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Coarse structure

01Definition

Intuition

02Coarse maps

03Examples

04Bounded sets

Sources and credits

Related topics

Bornology

Quasi-isometry

Uniform space