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One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?

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What is "topology compatible with the group operations" and why does the centralizer have to be closed? E.g., is the anti-discrete topology compatible? –  Alex Degtyarev Jan 22 at 10:39
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First: this only holds if the group is Hausdorff. Second: any set which is described by quant or-free equations in the language of groups. Third: is this a research-related question? –  doug Jan 22 at 10:40
    
@Alex: compatible means that the group is a topological group, i.e., the group operations are continuous. –  doug Jan 22 at 10:43
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Voting to reopen in light of Anton's nice answer. –  Benjamin Steinberg Jan 22 at 22:00
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Anton, yes, this is related to a research problem I am working on about the rigidity of the group topology on certain locally compact groups, and yes, you are right that I should have said Hausdorff. –  Rupert Jan 23 at 12:02

2 Answers 2

up vote 23 down vote accepted

Subsets of a group that are closed with respect to any Hausdorff group topology are called unconditionally closed.

Clearly, all algebraic sets are unconditionally closed, where a subset of a group $G$ is called algebraic if it is an intersection of finite unions of the sets of solutions to some equations with coefficient from $G$.

A.A.Markov proved that for countable groups the converse is also true: $$ \hbox{unconditionally closed = algebraic}. $$ For uncountable groups, this is not always the case as follows from works of S. Shelah (under CH) and G. Hesse.

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Klyachkon: Is there any English text about the work of Markov? Is there similar works for other algebraic systems? –  M. Shahryari Jan 23 at 4:03

This is not really an answer, rather a longer (and not very deep) remark about the question of the topology and the assumption that it should be Hausdorff.

Not every topology that makes a group a topological group is Hausdorff. But to any topological group we can associate a Hausdorff topological group in a canonical way. Let $G$ be a topological group and denote by $H$ the closure of $\{e\}$. Then $H$ is a normal subgroup in $G$, and the quotient group $G/H$ is Hausdorff with respect to the quotient topology. See Proposition 1-4 (vi) on page 6 of Ramakrishnan and Valenza, Fourier analysis on number field, 1999.

Based on this result, Ramakrishnan and Valenza write "Part (vi) shows that every topological group projects by a continuous homomorphism onto a topological group with Hausdorff topology. In this sense the assumption that a given group is Hausdorff is not too serious."

Nonetheless, the assumption plays an important role in Rupert's question. For, if we take the trivial topology $\mathcal{O}=\{\emptyset,G\}$, the $H=G$ and $G/H$ is the trivial group. Btw, the example of the trivial topology shows that the answer to the question (if we don't require Hausdorff) is easy: $G$ and the empty set are the only subsets that are closed in any topology on $G$ that makes $G$ a topological group.

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