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Vadim Alekseev
Vadim Alekseev
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I believe, this notion is closely related to the notion of the coarse structure which indeed had found many beautiful applications in geometric group theory and algebraic topology (including proofs of the Novikov conjecture for a lot of groups). For introduction, I'd recommend Lectures on coarse geometry by John Roe where in Chapter 2 he explains how to give a coarse structure to every metric space or a topological space (the latter upon fixing a compactification, which in your example should be Stone–Čech).

The idea of coarse geometry/topology is exactly dual to that of topology: one doesn't look at what happens at “small distances”, but rather at what happens at “large distances” in a space, and I believe, this is exactly what the definition of the cotopology also does (it defines what “large distances” mean).

However, the original definition of a coarse structure is done in a way as to define what are subsets with “not so large” distances, but I believe, it's a bit like defining topology by closed vs. open subsets or closure operation. I'll try to sketch below the translation between the two notions.

Definition. A coarse structure on a set $X$ is a collection $\mathcal E$ of subsets of $X \times X$ called controlled sets, so that $\mathcal E$ contains the identity relation, is closed under taking subsets, inverses (transposes), and finite unions, and is closed under composition of relations.

The main intuitive point in the connection can be already seen in the examples: the metric cotopological structure given by $R_r(x)$ should correspond to the family of controlled sets generated by $\{(x,y)\mid d(x,y)<r\}$.

Coarse to cotopological: Given a coarse structure $\mathcal E$, define a family of equivalence relations $\sim_{E,K}$, where $E$ runs through $\mathcal E$ and $K$ through subsets of $X$ in the following way: $y \sim_{E,K} z$ iff either $y=z$ or $(x,y),(x,z)\not\in E\cup E^{-1}$ for all $x\in K$. Observe that these equivalence relations are automatically symmetric, so it will be enough for us to work only with $E=E^{-1}\in\mathcal E$.

  • This family is trivially closed under arbitrary intersections: the intersection of $\sim_{E_i,K_i}$ is just $\sim_{\bigcap E_i,\bigcup K_i}$, and $\bigcap E_i\in \mathcal E$ because $\mathcal E$ is closed under taking subsets.
  • Finally, it is closed under passing to sus of two equivalence relations again using the properties of $\mathcal E$ as such as passing to subsets, finite unions and compositions (it's by no means obvious, but I think, I have figured out at least a sketch of a bit tedious proof; to not make the post too long, I omit it, but I'm happy to discuss this in detail if needed). I can sketch the proof in the metric case: The equivalence relation generated by $R_{r}(K)$ and $R_{r}(K')$ collapses everything exactly outside the ball of radius $r$ around $K\cap K'$, so it's just $R_{r}(K\cap K')$. If the radii are different, one has to work a bit more, but the flexibility of the coarse structure allows that.

Cotopological to coarse: It easy to see that an arbitrary intersection of coarse structures is again a coarse structure. Let's say that a coarse structure $\mathcal E$ is compatible with a family $\sim_\alpha$ of equivalence relations (i.e. a cotopology) if all $\sim_\alpha$ belong to the cotopology generated by $\mathcal E$. The intersection of all compatible coarse structures should then be the coarse structure which induces the given cotopology, although I haven't checked this carefully.

In any case, the possible connection here seems worth examining in detail.

I believe, this notion is closely related to the notion of the coarse structure which indeed had found many beautiful applications in geometric group theory and algebraic topology (including proofs of the Novikov conjecture for a lot of groups). For introduction, I'd recommend Lectures on coarse geometry by John Roe where in Chapter 2 he explains how to give a coarse structure to every metric space or a topological space (the latter upon fixing a compactification, which in your example should be Stone–Čech).

The idea of coarse geometry/topology is exactly dual to that of topology: one doesn't look at what happens at “small distances”, but rather at what happens at “large distances” in a space, and I believe, this is exactly what the definition of the cotopology also does (it defines what “large distances” mean).

However, the original definition of a coarse structure is done in a way as to define what are subsets with “not so large” distances, but I believe, it's a bit like defining topology by closed vs. open subsets. I'll try to sketch below the translation between the two notions.

Definition. A coarse structure on a set $X$ is a collection $\mathcal E$ of subsets of $X \times X$ called controlled sets, so that $\mathcal E$ contains the identity relation, is closed under taking subsets, inverses (transposes), and finite unions, and is closed under composition of relations.

The main intuitive point in the connection can be already seen in the examples: the metric cotopological structure given by $R_r(x)$ should correspond to the family of controlled sets generated by $\{(x,y)\mid d(x,y)<r\}$.

Coarse to cotopological: Given a coarse structure $\mathcal E$, define a family of equivalence relations $\sim_{E,K}$, where $E$ runs through $\mathcal E$ and $K$ through subsets of $X$ in the following way: $y \sim_{E,K} z$ iff either $y=z$ or $(x,y),(x,z)\not\in E\cup E^{-1}$ for all $x\in K$. Observe that these equivalence relations are automatically symmetric, so it will be enough for us to work only with $E=E^{-1}\in\mathcal E$.

  • This family is trivially closed under arbitrary intersections: the intersection of $\sim_{E_i,K_i}$ is just $\sim_{\bigcap E_i,\bigcup K_i}$, and $\bigcap E_i\in \mathcal E$ because $\mathcal E$ is closed under taking subsets.
  • Finally, it is closed under passing to sus of two equivalence relations again using the properties of $\mathcal E$ as such as passing to subsets, finite unions and compositions (it's by no means obvious, but I think, I have figured out at least a sketch of a bit tedious proof; to not make the post too long, I omit it, but I'm happy to discuss this in detail if needed). I can sketch the proof in the metric case: The equivalence relation generated by $R_{r}(K)$ and $R_{r}(K')$ collapses everything exactly outside the ball of radius $r$ around $K\cap K'$, so it's just $R_{r}(K\cap K')$. If the radii are different, one has to work a bit more, but the flexibility of the coarse structure allows that.

Cotopological to coarse: It easy to see that an arbitrary intersection of coarse structures is again a coarse structure. Let's say that a coarse structure $\mathcal E$ is compatible with a family $\sim_\alpha$ of equivalence relations (i.e. a cotopology) if all $\sim_\alpha$ belong to the cotopology generated by $\mathcal E$. The intersection of all compatible coarse structures should then be the coarse structure which induces the given cotopology, although I haven't checked this carefully.

In any case, the possible connection here seems worth examining in detail.

I believe, this notion is closely related to the notion of the coarse structure which indeed had found many beautiful applications in geometric group theory and algebraic topology (including proofs of the Novikov conjecture for a lot of groups). For introduction, I'd recommend Lectures on coarse geometry by John Roe where in Chapter 2 he explains how to give a coarse structure to every metric space or a topological space (the latter upon fixing a compactification, which in your example should be Stone–Čech).

The idea of coarse geometry/topology is exactly dual to that of topology: one doesn't look at what happens at “small distances”, but rather at what happens at “large distances” in a space, and I believe, this is exactly what the definition of the cotopology also does (it defines what “large distances” mean).

However, the original definition of a coarse structure is done in a way as to define what are subsets with “not so large” distances, but I believe, it's a bit like defining topology by open subsets or closure operation. I'll try to sketch below the translation between the two notions.

Definition. A coarse structure on a set $X$ is a collection $\mathcal E$ of subsets of $X \times X$ called controlled sets, so that $\mathcal E$ contains the identity relation, is closed under taking subsets, inverses (transposes), and finite unions, and is closed under composition of relations.

The main intuitive point in the connection can be already seen in the examples: the metric cotopological structure given by $R_r(x)$ should correspond to the family of controlled sets generated by $\{(x,y)\mid d(x,y)<r\}$.

Coarse to cotopological: Given a coarse structure $\mathcal E$, define a family of equivalence relations $\sim_{E,K}$, where $E$ runs through $\mathcal E$ and $K$ through subsets of $X$ in the following way: $y \sim_{E,K} z$ iff either $y=z$ or $(x,y),(x,z)\not\in E\cup E^{-1}$ for all $x\in K$. Observe that these equivalence relations are automatically symmetric, so it will be enough for us to work only with $E=E^{-1}\in\mathcal E$.

  • This family is trivially closed under arbitrary intersections: the intersection of $\sim_{E_i,K_i}$ is just $\sim_{\bigcap E_i,\bigcup K_i}$, and $\bigcap E_i\in \mathcal E$ because $\mathcal E$ is closed under taking subsets.
  • Finally, it is closed under passing to sus of two equivalence relations again using the properties of $\mathcal E$ as such as passing to subsets, finite unions and compositions (it's by no means obvious, but I think, I have figured out at least a sketch of a bit tedious proof; to not make the post too long, I omit it, but I'm happy to discuss this in detail if needed). I can sketch the proof in the metric case: The equivalence relation generated by $R_{r}(K)$ and $R_{r}(K')$ collapses everything exactly outside the ball of radius $r$ around $K\cap K'$, so it's just $R_{r}(K\cap K')$. If the radii are different, one has to work a bit more, but the flexibility of the coarse structure allows that.

Cotopological to coarse: It easy to see that an arbitrary intersection of coarse structures is again a coarse structure. Let's say that a coarse structure $\mathcal E$ is compatible with a family $\sim_\alpha$ of equivalence relations (i.e. a cotopology) if all $\sim_\alpha$ belong to the cotopology generated by $\mathcal E$. The intersection of all compatible coarse structures should then be the coarse structure which induces the given cotopology, although I haven't checked this carefully.

In any case, the possible connection here seems worth examining in detail.

Source Link
Vadim Alekseev
Vadim Alekseev
  • 1.5k
  • 11
  • 16

I believe, this notion is closely related to the notion of the coarse structure which indeed had found many beautiful applications in geometric group theory and algebraic topology (including proofs of the Novikov conjecture for a lot of groups). For introduction, I'd recommend Lectures on coarse geometry by John Roe where in Chapter 2 he explains how to give a coarse structure to every metric space or a topological space (the latter upon fixing a compactification, which in your example should be Stone–Čech).

The idea of coarse geometry/topology is exactly dual to that of topology: one doesn't look at what happens at “small distances”, but rather at what happens at “large distances” in a space, and I believe, this is exactly what the definition of the cotopology also does (it defines what “large distances” mean).

However, the original definition of a coarse structure is done in a way as to define what are subsets with “not so large” distances, but I believe, it's a bit like defining topology by closed vs. open subsets. I'll try to sketch below the translation between the two notions.

Definition. A coarse structure on a set $X$ is a collection $\mathcal E$ of subsets of $X \times X$ called controlled sets, so that $\mathcal E$ contains the identity relation, is closed under taking subsets, inverses (transposes), and finite unions, and is closed under composition of relations.

The main intuitive point in the connection can be already seen in the examples: the metric cotopological structure given by $R_r(x)$ should correspond to the family of controlled sets generated by $\{(x,y)\mid d(x,y)<r\}$.

Coarse to cotopological: Given a coarse structure $\mathcal E$, define a family of equivalence relations $\sim_{E,K}$, where $E$ runs through $\mathcal E$ and $K$ through subsets of $X$ in the following way: $y \sim_{E,K} z$ iff either $y=z$ or $(x,y),(x,z)\not\in E\cup E^{-1}$ for all $x\in K$. Observe that these equivalence relations are automatically symmetric, so it will be enough for us to work only with $E=E^{-1}\in\mathcal E$.

  • This family is trivially closed under arbitrary intersections: the intersection of $\sim_{E_i,K_i}$ is just $\sim_{\bigcap E_i,\bigcup K_i}$, and $\bigcap E_i\in \mathcal E$ because $\mathcal E$ is closed under taking subsets.
  • Finally, it is closed under passing to sus of two equivalence relations again using the properties of $\mathcal E$ as such as passing to subsets, finite unions and compositions (it's by no means obvious, but I think, I have figured out at least a sketch of a bit tedious proof; to not make the post too long, I omit it, but I'm happy to discuss this in detail if needed). I can sketch the proof in the metric case: The equivalence relation generated by $R_{r}(K)$ and $R_{r}(K')$ collapses everything exactly outside the ball of radius $r$ around $K\cap K'$, so it's just $R_{r}(K\cap K')$. If the radii are different, one has to work a bit more, but the flexibility of the coarse structure allows that.

Cotopological to coarse: It easy to see that an arbitrary intersection of coarse structures is again a coarse structure. Let's say that a coarse structure $\mathcal E$ is compatible with a family $\sim_\alpha$ of equivalence relations (i.e. a cotopology) if all $\sim_\alpha$ belong to the cotopology generated by $\mathcal E$. The intersection of all compatible coarse structures should then be the coarse structure which induces the given cotopology, although I haven't checked this carefully.

In any case, the possible connection here seems worth examining in detail.