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Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is complete.

Question 1. Does there exist a complete topologizable group?

In particular:

Question 2. Is the group $SO(3,\mathbb R)$ complete?

Question 3. Is the group $Sym(\mathbb N)$ complete?

A simple Baire category argument shows that each complete topologizable group is uncountable.

Remark. There many examples of Polish groups admitting a unique $\omega$-narrow Hausdorff group topology (so, each $\omega$-narrow Hausdorff group topology on such a group is complete), see http://www.math.uiuc.edu/~ssolecki/papers/AutomaticContinuity13.pdf.

Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is complete.

Question 1. Does there exist a complete topologizable group?

In particular:

Question 2. Is the group $SO(3,\mathbb R)$ complete?

Question 3. Is the group $Sym(\mathbb N)$ complete?

A simple Baire category argument shows that each complete topologizable group is uncountable.

Remark. There are many examples of Polish groups admitting a unique $\omega$-narrow Hausdorff group topology (so, each $\omega$-narrow Hausdorff group topology on such a group is complete), see http://www.math.uiuc.edu/~ssolecki/papers/AutomaticContinuity13.pdf.
In particular, $Sym(\mathbb N)$ is such a group.

Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is complete.

Question 1. Does there exist a complete topologizable group?

In particular:

Question 2. Is the group $SO(3,\mathbb R)$ complete?

A simple Baire category argument shows that each complete topologizable group is uncountable.

Remark. There many examples of Polish groups admitting a unique $\omega$-narrow Hausdorff group topology (so, each $\omega$-narrow Hausdorff group topology on such a group is complete), see http://www.math.uiuc.edu/~ssolecki/papers/AutomaticContinuity13.pdf.

Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is complete.

Question 1. Does there exist a complete topologizable group?

In particular:

Question 2. Is the group $SO(3,\mathbb R)$ complete?

Question 3. Is the group $Sym(\mathbb N)$ complete?

A simple Baire category argument shows that each complete topologizable group is uncountable.

Remark. There many examples of Polish groups admitting a unique $\omega$-narrow Hausdorff group topology (so, each $\omega$-narrow Hausdorff group topology on such a group is complete), see http://www.math.uiuc.edu/~ssolecki/papers/AutomaticContinuity13.pdf.