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Noah Schweber
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This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing answers were aimed at the earlier version of the question.


Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow\mathbb{R}$ with the compact-open topology, and consider the following two subspaces:

  • $\mathcal{Asso}$ = the subspace of all associative functions.

  • $\mathcal{Comm}$ = the subspace of all commutative functions.

Of course these are distinct as subsets; my question is whether they are topologically distinguishable:

Is $\mathcal{Asso}\cong\mathcal{Comm}$?

I strongly suspect that the answer is negative - in particular, $\mathcal{Comm}$ is connected and I suspect $\mathcal{Asso}$ is not - but I don't see how to prove this. (Another nice feature of $\mathcal{Comm}$ is that it is an $\mathbb{R}$-vector space in a natural way; however, it's not clear how to extract any power from this observation.)

Generally, $\mathcal{Asso}$ seems rather mysterious, and I'm interested in any information about it.

This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context.


Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow\mathbb{R}$ with the compact-open topology, and consider the following two subspaces:

  • $\mathcal{Asso}$ = the subspace of all associative functions.

  • $\mathcal{Comm}$ = the subspace of all commutative functions.

Of course these are distinct as subsets; my question is whether they are topologically distinguishable:

Is $\mathcal{Asso}\cong\mathcal{Comm}$?

I strongly suspect that the answer is negative - in particular, $\mathcal{Comm}$ is connected and I suspect $\mathcal{Asso}$ is not - but I don't see how to prove this. (Another nice feature of $\mathcal{Comm}$ is that it is an $\mathbb{R}$-vector space in a natural way; however, it's not clear how to extract any power from this observation.)

Generally, $\mathcal{Asso}$ seems rather mysterious, and I'm interested in any information about it.

This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing answers were aimed at the earlier version of the question.


Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow\mathbb{R}$ with the compact-open topology, and consider the following two subspaces:

  • $\mathcal{Asso}$ = the subspace of all associative functions.

  • $\mathcal{Comm}$ = the subspace of all commutative functions.

Of course these are distinct as subsets; my question is whether they are topologically distinguishable:

Is $\mathcal{Asso}\cong\mathcal{Comm}$?

I strongly suspect that the answer is negative - in particular, $\mathcal{Comm}$ is connected and I suspect $\mathcal{Asso}$ is not - but I don't see how to prove this. (Another nice feature of $\mathcal{Comm}$ is that it is an $\mathbb{R}$-vector space in a natural way; however, it's not clear how to extract any power from this observation.)

Generally, $\mathcal{Asso}$ seems rather mysterious, and I'm interested in any information about it.

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Noah Schweber
  • 20.7k
  • 10
  • 111
  • 332

This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context.


Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$$\mathbb{R}^2\rightarrow\mathbb{R}$ with the compact-open topology. I'm interested in analyzing subspaces of $C(\mathbb{R}^2,\mathbb{R})$ determined by first-order theories, especially finite equational theories. Specifically, for $T$ a set of first-order sentences in the language of a single binary operation, let $T_\mathbb{R}$ be the subspace of continuous $f$ such that $(\mathbb{R};f)\models T$. I'm curious about how $T$ affectsand consider the purely topological properties of $T_\mathbb{R}$. For example, when is $T_\mathbb{R}$ connected? etc.following two subspaces:

  • $\mathcal{Asso}$ = the subspace of all associative functions.

  • $\mathcal{Comm}$ = the subspace of all commutative functions.

However, even in very concrete cases, I'm having trouble understanding what $T_\mathbb{R}$ looks like. Letting $C$ and $A$ be the usual statements of commutativity and associativity, I think an answer to the followingOf course these are distinct as subsets; my question would clear things up immenselyis whether they are topologically distinguishable:

Question: is Is $\{C\}_\mathbb{R}\cong\{A\}_\mathbb{R}$$\mathcal{Asso}\cong\mathcal{Comm}$?

$\{C\}_\mathbb{R}$ I strongly suspect that the answer is pretty tame since as Eric Wofsey observed it's a vector subspace ofnegative $C(\mathbb{R}^2,\mathbb{R})$. However- in particular, $\{A\}_\mathbb{R}$ seems much weirder. For example,$\mathcal{Comm}$ is connected and I suspect $\mathcal{Asso}$ is not - but I don't even know whethersee how to prove this. $\{A\}_\mathbb{R}$(Another nice feature of $\mathcal{Comm}$ is connectedthat it is an $\mathbb{R}$-vector space in a natural way; however, it's not clear how to extract any power from this observation.)

Generally, $\mathcal{Asso}$ seems rather mysterious, and I'm interested in any information about it.

This was previously asked and bountied at math.stackexchange with no response.


Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$ with the compact-open topology. I'm interested in analyzing subspaces of $C(\mathbb{R}^2,\mathbb{R})$ determined by first-order theories, especially finite equational theories. Specifically, for $T$ a set of first-order sentences in the language of a single binary operation, let $T_\mathbb{R}$ be the subspace of continuous $f$ such that $(\mathbb{R};f)\models T$. I'm curious about how $T$ affects the purely topological properties of $T_\mathbb{R}$. For example, when is $T_\mathbb{R}$ connected? etc.

However, even in very concrete cases, I'm having trouble understanding what $T_\mathbb{R}$ looks like. Letting $C$ and $A$ be the usual statements of commutativity and associativity, I think an answer to the following question would clear things up immensely:

Question: is $\{C\}_\mathbb{R}\cong\{A\}_\mathbb{R}$?

$\{C\}_\mathbb{R}$ is pretty tame since as Eric Wofsey observed it's a vector subspace of $C(\mathbb{R}^2,\mathbb{R})$. However, $\{A\}_\mathbb{R}$ seems much weirder. For example, I don't even know whether $\{A\}_\mathbb{R}$ is connected.

This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context.


Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow\mathbb{R}$ with the compact-open topology, and consider the following two subspaces:

  • $\mathcal{Asso}$ = the subspace of all associative functions.

  • $\mathcal{Comm}$ = the subspace of all commutative functions.

Of course these are distinct as subsets; my question is whether they are topologically distinguishable:

Is $\mathcal{Asso}\cong\mathcal{Comm}$?

I strongly suspect that the answer is negative - in particular, $\mathcal{Comm}$ is connected and I suspect $\mathcal{Asso}$ is not - but I don't see how to prove this. (Another nice feature of $\mathcal{Comm}$ is that it is an $\mathbb{R}$-vector space in a natural way; however, it's not clear how to extract any power from this observation.)

Generally, $\mathcal{Asso}$ seems rather mysterious, and I'm interested in any information about it.

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Noah Schweber
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Comparing "axiomatized function spaces"

This was previously asked and bountied at math.stackexchange with no response.


Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$ with the compact-open topology. I'm interested in analyzing subspaces of $C(\mathbb{R}^2,\mathbb{R})$ determined by first-order theories, especially finite equational theories. Specifically, for $T$ a set of first-order sentences in the language of a single binary operation, let $T_\mathbb{R}$ be the subspace of continuous $f$ such that $(\mathbb{R};f)\models T$. I'm curious about how $T$ affects the purely topological properties of $T_\mathbb{R}$. For example, when is $T_\mathbb{R}$ connected? etc.

However, even in very concrete cases, I'm having trouble understanding what $T_\mathbb{R}$ looks like. Letting $C$ and $A$ be the usual statements of commutativity and associativity, I think an answer to the following question would clear things up immensely:

Question: is $\{C\}_\mathbb{R}\cong\{A\}_\mathbb{R}$?

$\{C\}_\mathbb{R}$ is pretty tame since as Eric Wofsey observed it's a vector subspace of $C(\mathbb{R}^2,\mathbb{R})$. However, $\{A\}_\mathbb{R}$ seems much weirder. For example, I don't even know whether $\{A\}_\mathbb{R}$ is connected.